Summary of studies used in the analysis.
\r\n\tIt is a relatively simple process and a standard tool in any industry. Because of the versatility of the titration techniques, nearly all aspects of society depend on various forms of titration to analyze key chemical compounds.
\r\n\tThe aims of this book is to provide the reader with an up-to-date coverage of experimental and theoretical aspects related to titration techniques used in environmental, pharmaceutical, biomedical and food sciences.
Breast cancer is the most common cancer among women, accounting for one-third of cancers diagnosed. Statistically, ~230,000 new cases of invasive breast cancer and ~63,000 in situ breast carcinomas are diagnosed in the US annually; breast cancer is the third leading cause of cancer death among women, causing ~40,000 deaths in the US every year [1]. According to SEER statistics, approximately 61% of women are found to have localized breast cancers at the time of diagnosis; about 31% are found to be regional disease; another 5% are diagnosed with distant metastases while about 3% are unstaged [2]. The 5-year survival rate for women with localized cancer is 98%; for those with regional disease, it drops to 84%; for those diagnosed with distant stage, the survival rate drops dramatically to 23%; while for unstaged cancers the 5-year survival rate is about 58%. Figure 1 illustrates the dependence of survival on cancer stage.
\nThe dependence of mortality rates on cancer type and stage. From Kerlikowske et al. [3].
There are many reasons why cancers are not detected early but some of the major factors relate to limited participation in breast screening and the performance of screening mammography.
\nNational cancer screening statistics indicate that only 51% of eligible women undergo annual mammograms [4]. That rate is even lower for African American women and/or those of lower socioeconomic groups. Access, fear of radiation and discomfort are some of the factors that contribute to the low participation rate. Greater participation would lead to detection of breast cancer at an earlier stage leading to longer survival. Increased participation and improved breast cancer detection would have the greatest effect on the statistic of nearly 1 in 3 women who are diagnosed each year with later stage (regional or greater) breast cancer, totaling approximately 60,000 women per year in the USA. The net effect would be an increase in survival time and a corresponding decrease in mortality rates. This is also suggested in a recent meta-analysis, whereby increased participation and sensitivity lead to additional invasive cancer detection and greater mortality reduction [4].
\nFor women with dense breast tissue, who are at the highest risk for developing breast cancer [5–8], the performance of mammography is at its worst [9]. Consequently, many cancers are missed at their earliest stages when they are the most treatable. Improved cancer detection for women with denser breasts would decrease the proportion of breast cancers diagnosed at later stages, which would significantly lower the mortality rate.
\nX-ray mammography detects about 5 cancers per 1000 screens [10]. However, its positive predictive value (PPV) is low and its sensitivity is greatly reduced in women with dense breast tissue [10]. Although digital breast tomosynthesis (DBT) may improve upon some of the limitations of standard mammography, it is unlikely to create a paradigm shift in performance [11] while generating even higher levels of ionizing radiation [12]. MRI can significantly improve on these limitations by virtue of its volumetric, radiation-free imaging capability. Studies have shown that MRI can have a positive impact in the breast management continuum ranging from risk assessment to diagnosis and treatment monitoring [12, 13]. However, MRI can have a high false positive rate, requires contrast injection and the exams can be both long and costly [14]. Furthermore, MR has long been prohibitively expensive for routine use and there is a need for a low-cost equivalent alternative. Yet, for high-risk women, MRI is now viewed as the gold standard for breast cancer detection and screening [15–23]. Positron emission tomography is also limited by cost and radiation concerns.
\nRecent studies have demonstrated the effectiveness of hand held ultrasound imaging in detecting breast cancer, particularly for women with dense breasts (Table 1). These studies have shown that up to 4.5 extra cancers were detected per 1000 screens [24–34]. A striking aspect of the added detections is that they are predominantly node negative invasive cancers which would have potentially progressed to a later stage before possible mammographic detection. Moreover, there is little risk of over detection of ductal carcinoma in situ (DCIS). The sensitivity of mammography is greater for DCIS than it is for invasive cancer, with DCIS making up approximately 25% of mammographic screen-detected breast cancers [35].
\nAuthor (Year) | \nCenter | \nType | \nExams | \nUS only cancers | \nYield per 1000 | \n
---|---|---|---|---|---|
Brem et al. (2014) | \nMulti | \nABUS | \n15,318 | \n30 | \n1.96 | \n
Berg et al. (2012) | \nMulti | \nHHUS | \n7473 | \n32 | \n4.28 | \n
Hooley et al. (2012) | \nSingle | \nHHUS | \n935 | \n3 | \n3.21 | \n
Kelly et al. (2010) | \nMulti | \nAWBU | \n6425 | \n23 | \n3.58 | \n
Corsetti et al. (2008) | \nMulti | \nHHUS | \n9157 | \n37 | \n4.04 | \n
Crystal et al. (2003) | \nSingle | \nHHUS | \n1517 | \n7 | \n4.61 | \n
Leconte et al. (2003) | \nSingle | \nHHUS | \n4236 | \n16 | \n3.78 | \n
Kolb et al. (2002) | \nSingle | \nHHUS | \n13,547 | \n37 | \n2.73 | \n
Kaplan (2001) | \nSingle | \nHHUS | \n1862 | \n6 | \n3.22 | \n
Buchberger et al. (2000) | \nSingle | \nHHUS | \n8103 | \n32 | \n3.95 | \n
Gordon et al. (1995) | \nSingle | \nHHUS | \n12,706 | \n44 | \n3.46 | \n
Summary of studies used in the analysis.
We have examined the data from these studies to extract the statistics of cancer detection by imaging mode (Table 1). The results are summarized in Figure 2. It is striking to note that ultrasound (US) almost doubles the cancer detection rate in dense breasts. However, despite these successful study outcomes, handheld ultrasound is unlikely to be adopted for screening because it is operator dependent, and its imaging aperture is small, which hinders whole breast imaging. Furthermore, ultrasound’s increased sensitivity to invasive cancer is offset by lowered sensitivity to DCIS by virtue of mammography’s greater ability to detect microcalcifications. Although such a trade-off may be justified by the fact that mortality from invasive cancers is much higher than that from DCIS, a combined screening [mammography plus automated breast ultrasound (ABUS)] would provide a comprehensive screen. It has therefore been proposed that ABUS be used for screening, supplemental to mammography.
\nVenn diagram summarizing comparative cancer detection rates for screening mammography and ultrasound.
To that end, automated breast ultrasound (ABUS) has been introduced as a way of overcoming these issues, mainly by reducing operator dependence and increasing the field of view. For example, the GE Invenia ABUS ultrasound system for breast cancer screening, originally developed by U-Systems., recently received screening approval, adjunctive to mammography, from the FDA, because it demonstrated an ability to detect cancers missed by mammography in dense breasts. The SomoInsight screening study [24], indeed showed that ABUS plus mammography outperformed mammography alone, leading to the first FDA approval for ultrasound screening for breast cancer.
\nThe fundamental quandary of breast screening today is the knowledge that (i) mammography misses cancers in dense breasts, (ii) that Automated Breast ultrasound (ABUS) detects cancers that mammography misses and yet (iii) screening continues largely with mammography only. This paradox is amplified even further by the proliferation of state breast density notification laws in the USA which mandate that this information be available to women undergoing breast cancer screening. The primary reason this paradox exists today is that ABUS screening increases call back rates (up to a factor of two in case of the SomoInsight study [23]). The improvement in classification performance, measured by the area under the ROC curve, is modest because the increase in sensitivity is partially offset by an increase in false positives thus slowing its adoption. Technically, with its basic B-mode capability, ABUS has the same issue with false positives as hand held ultrasound. It is therefore unlikely that ABUS will be widely adopted for screening in the foreseeable future without more tissue-specific imaging capability. Improved lesion characterization would help lower the barriers to adoption of screening ultrasound.
\nUltrasound tomography (UST) is an emerging technique that has the potential for tissue-specific imaging and characterization, by virtue of its transmission imaging capability [36–61]. Improved specificity would lower call back rates and lower the barriers to adoption. An adjunctive use of UST would have the potential to improve specificity relative to current ABUS and provide a comprehensive screen that would uncover invasive cancers otherwise missed by mammography. Detection of such early stage invasive cancers would provide women with curative treatment, the opportunity for which might be otherwise lost.
\nConventional reflection ultrasound exploits differences in acoustic impedance between tissue types to provide anatomical images of breast tumors [62, 63]. However, reflection is just one aspect of a multi-faceted set of acoustic signatures associated with the biomechanical properties of tissue. UST is a technique that moves beyond B-mode imaging by virtue of its transmission capabilities. The latter provides additional characterization by measuring tissue parameters such as sound speed and attenuation (ATT) [64–68]. These parameters can be used to characterize lesions in a quantitative manner, a capability not available in current whole breast ultrasound systems. By merging reflection images with images of the bio-acoustic parameters of sound speed and attenuation, UST offers the possibility of exploiting differences in anatomical and physical properties of tissue to accurately differentiate cancer from normal tissue or benign disease. UST parameters are also quantitative, which allows new consideration of second and third-order statistical image analyses, or radiomics. Ultrasound has previously not been suitable for the burgeoning applications of radiomics due to its lack of true quantitative parameters such as sound speed (m/s) and attenuation (dB/cm/MHz). Initial assessments of UST performance was carried out, as described below.
\nIn an initial attempt to assess the potential of UST in breast imaging, studies were carried out at the Karmanos Cancer Institute, Detroit, MI, USA. Informed consent was obtained from all patients, prospectively recruited in an IRB-approved protocol following HIPAA guidelines. Patients were scanned at the Alexander J Walt Comprehensive Breast Center. Standard multi-modality imaging was available for all patients. The Walt Breast Center houses SoftVue, a UST system manufactured by Delphinus Medical Technologies, Inc (Novi, MI). SoftVue embodies a number of attributes that differentiate it from conventional imaging modalities:
Water-based pulse coupling: SoftVue utilizes a water filled imaging chamber that is kept at body temperature. Its primary purpose is to couple the sound energy between the transducer and the breast tissue.
Closed geometry probe: A circular ring transducer surrounds the breast while both are immersed in water. There is no compression of the breast since the transducer is offset from the breast with water acting as the pulse coupling agent. The closed transducer geometry allows collection of signals that pass through the entire width of the breast, a requirement for transmission imaging and the reconstruction of sound speed and attenuation images. These parameters provide quantitative information in absolute units that are tied to externals standards (km/s and dB/cm, respectively).
Operator independence: Unlike mammography and other ABUS systems, multiple positionings are not required for larger breasts. Once the patient is positioned on the table, the operator simply presses the button and the exam is performed automatically without further intervention from the operator.
Scan time: SoftVue scan time is 1–2 min per breast (depending on breast size). This scan duration minimizes intra-slice and inter-slice motion artifacts.
Image reconstruction time. In this study, reconstruction time for a bilateral breast exam was ~30 min for the average patient and current hardware/software processing ability.
SoftVue was used to scan the recruited patients for this study. Coronal image series were produced by tomographic algorithms for reflection, sound speed and attenuation. All images were reviewed by a board-certified radiologist who has more than 20 years of experience in breast imaging and US-technology development. Symptomatic study participants were scanned with a SoftVue UST system. Pathological correlation was based on biopsy results and standard imaging (e.g. US definitive cyst).
\nTomographic algorithms were used to generate images stacks of reflectivity, sound speed and attenuation for each patient. Lesions were identified based on correlation with standard imaging so that the tumor sound speed (SS) and attenuation (ATT) could be assessed. An example each type of image is shown in Figure 3.
\nFrom left to right, reflection, sound speed and attenuation image slices depicting breast parenchyma and a fibroadenoma at 7 o’clock.
In the first phase of the study, correlative comparisons with multi-modal imaging were carried out to assess lesion properties relative to mammography, US and MR. In the second phase, MR breast imaging was used to establish basic operational capabilities of the UST system including the identification and characterization of parenchymal patterns, determination of the spatial resolution of UST and an estimate the breast volume that can imaged with UST. The third phase of the study focused on lesion characterization. Region of interest (ROI) analyses were performed on all identified lesions using all three UST image types. Combinations of the ROI generated quantitative values were used to characterize all masses, particularly in relation to relative differences with surrounding peritumoral regions.
\nSince the patients were recruited at KCI on the basis of having a suspicious finding, standard imaging such as mammography, US and sometimes MRI were available, as well as the radiology and pathology reports. These images and the associated reports were used to retroactively locate the lesions in the UST image stacks for visual comparison. Figures 4–7 show examples of UST images in relation to the other modalities. When MRI was available, the images were projected into the coronal plane for easier comparison with the UST whose native format is coronal.
\nA 9 mm IDC at 3 o’clock. CC and MLO mammographic views of the affected breast are shown on the left with the lesion identified by arrows. The coronal UST views are shown in the form of reflection, sound speed and attenuation images. The corresponding ultrasound and MR images are also shown.
Multimodality images compared to UST reflection, sound speed and attenuation. An IDC is shown at 12 o’clock.
Multimodality images vs UST reflection, sound speed and attenuation showing an IDC and intramammary lymph node.
Illustrating the chest wall access achievable by UST relative to mammography.
Figure 4 shows a 9mm IDC at 3 o’clock. CC and MLO mammographic views of the affected breast are shown on the left with the lesion identified by arrows. The UST views corresponding to the coronal planes that contain the lesions are across the top with reflection, sound speed and attenuation images laid out from left to right. The corresponding ultrasound and MR images are shown along the bottom. Inspection of the images shows good correspondence in shape and location of the lesion. The greatest similarity is between the UST images and MRI. The IDC is seen to be hypoechoic in reflection and has high sound speed and attenuation contrast. An IDC in a heterogeneously dense breast is shown in Figure 5 This IDC was initially missed by mammography. A large IDC and an intramammary lymph node are shown in Figure 6. Note the concordance between the UST images and mammography. Figure 7 illustrates the chest wall access achievable by UST relative to mammography. Although UST does not access the entire axilla it does visualize the cancer that has invaded the chest wall.
\nUST and MR imaging was performed within weeks of each other. UST imaging was carried out with the SoftVue system (Delphinus Medical Technologies) and the MR exams with a Philips Achieva 3T system. The resulting image sequences were qualitatively and quantitatively to assess imaging performance of UST. As discussed above, UST images correlate best with MR images. Further inspection shows that of the three UST image types, the sound speed image correlates best with MR. Figure 8 shows a coronal view comparison between UST speed of sound and MR contrast-enhanced fat subtracted images of representative breast parenchyma.
\nTop: Coronal UST sound speed images for six different patients. Bottom: Corresponding fat subtracted contrast-enhanced MR images.
The parenchymal patterns are very similar with the only major difference relating to the shape of the breast. This difference can be explained by the fact that the SoftVue system utilizes water so that buoyancy foreshortens the breast while with MR, gravity lengthens the breast in the AP dimension (i.e. prone).
\nAs discussed above, UST images correlate best with MR images. Further inspection shows that of the three UST image types, the sound speed image correlates best with MR, as illustrated in Figure 8. The parenchymal patterns are very similar with the only major difference relating to the shape of the breast. This difference can be explained by the fact that the SoftVue system utilizes water so that the buoyancy force helps shape the breast while with MR, gravity shapes the breast.
\nMRI was used as the gold standard for defining the extent of the breast tissue. MRI and UST breast volumes were compared using a paired t-test. In the first step, a k-means segmentation algorithm was applied to T1 breast MR images to automatically separate out the non-tissue background. In the second step, the boundary between the breast tissue and the chest wall was drawn manually and the chest wall removed, leaving behind only breast tissue (Figure 9).
\nThe segmentation process for MR images (top) and UST images (bottom). From left to right, original image, segmentation boundary and the final segmented image.
In the UST images a semi-automated tool was used to draw a boundary around the breast tissue in each coronal slice and everything outside the boundary removed (water signal). Any slices containing chest wall signal were also removed. The resulting stack of slices then represented the pure breast volume scanned by UST.
\nThe two sets of volumes were plotted against each other as shown in Figure 10. The average breast volumes for MRI and UST were compared and the result shown in Table 2. As expected, the UST scanned volume was less than that of MRI and was found to be about 89% of the MRI volume on average. However, a student’s paired t-test indicates that this difference is not significant. Since UST cannot fully access the axilla, it is likely that the UST scanned volume is somewhat lower than that of MRI, even though UST generally reaches the pectoralis muscle at the chest wall.
\nCorrelation between UST and MR measured breast volumes.
Mean MRI volume (cm3) | \nMean UST volume (cm3) | \np Value | \n
---|---|---|
1224 | \n1089 | \n0.113 | \n
Volume comparison.
The spatial resolution of each modality was estimated using profile cuts of thin features using, the full-width, half-maximum criterion as shown in Figure 11. The results of the spatial resolution analysis are shown in Table 3. The spatial resolution was found to be dependent on the reprojection type for both MRI and with UST outperforming MRI in the coronal plane and MRI outperforming UST in the other projections. (However, MR acquisitions with isotropic voxels would show comparable resolution to UST in the coronal plane). The UST image voxels are not isotropic and data acquisition cannot be readily adjusted like MR, such that UST reconstructed in axial and sagittal planes have resolution that approach the 2.5 mm slice thickness at this time.
\nThe spatial resolution of each modality was estimated using profile cuts of thin features using, the full-width, half-maximum criterion, as illustrated.
Resolution | \nUST | \nMRI | \n
---|---|---|
Coronal | \n0.7 ± 0.1 mm | \n1.6 ± 0.3 mm | \n
Axial/sagittal | \n2.5 ± 0.5 mm | \n0.8 ± 0.1 mm | \n
Spatial resolution comparison.
Ultrasound breast imaging reporting and data system (US-BI-RADS) criteria are predominantly devoted to assessment of tumor shape, margins and interaction with adjacent tissue. However, criteria such as shadowing or enhanced through transmission are not applicable to UST’s circular geometry. In addition, UST, operating at 3 MHz, appears more sensitive to the specular reflectors of benign mass capsules, or the spiculations and/or architectural distortions of many cancers. Therefore, we developed a 5-point scale that combined US-BI-RADS criteria for tumor margins, as well as possibilities for peritumoral tissue interaction (Figure 12).
\nSchematic of shape and margin analysis and associated grading scheme.
Masses were characterized by a (i) Margin Boundary score, (ii) reflectivity, (iii) quantitative SS evaluation and (iv) ATT evaluations. A semi-automatic region-of-interest (ROI) tool was used to determine the quantitative properties of each mass. After identifying the mass of interest, a simple elliptical ROI is drawn around the mass. The ROI algorithm then generates 20 radial ellipsoids – 10 inside and 10 outside the mass. Quantitative information was then measured for each of the 20 annuli for subsequent analysis. The region of interest (ROI) analysis was performed on all identified lesions using all three UST image types. Combinations of the ROI generated values were used to characterize all masses in the study.
\nOngoing analyses of the ROI tool have not yet led to full evaluation of second and third-order statistics of textural analyses, as well as their impacts upon decision analysis and predictive values. However, our recent RSNA presentation highlighted the significant impacts of first-order statistics such as standard deviation, within the tumoral ROI and comparisons with the surrounding peritumoral region [69]. Scatterplots and box plots of the optimal methods were used to illustrate the characterization potential. The box plot in Figure 13 shows the differentiation achieved when using the boundary score (Figure 6) combined with the first-order statistic of standard deviation, a more crude measure of heterogeneity, based upon tumoral ROI extracted from ATT images, which had only slightly higher significance than SS [69]. These ROIs were again obtained by simply drawing an elliptical ROI around the mass and determining the standard deviation with in the ROI. The box plot was based on taking the average values for 107 benign lesions and 31 cancers [69].
\nSeparation of cancer from benign when using boundary score and hetrogeneity score.
Upon further investigation, it was found that the SS of the peritumoral mass region (defined by an annular area just outside the mass boundary ROI) further separated the benign masses from cancer. A scatter plot based on all of these parameters is shown in Figure 14. The scatter plot shows separately the cancers, fibroadenomas and cancers. The cancers are tightly grouped in the top left corner of the plot indicating high boundary scores, high heterogeneity and lower peritumoral sound speed. By these measures, there was not much separation between cysts and fibroadenomas but significant separation between them and cancer. ROC analysis of the data represented in the scatter plot indicates a PPV of 91% when the sensitivity is 97%. However, this is a subset of data relative to an expanded ongoing study that includes more quantitative margin analyses. The ultimate goal is to generate textural analyses that will be less operator dependent and serve as appropriate diagnostic aids for a detected mass by simply requiring the radiologist to draw an ellipsoidal ROI. This method can also serve as a teaching tool for identifying grossly apparent textural differences within the tumor and surrounding peritumoral region. Figure 15 shows the basic differences in sound speed texture noted for many cysts, fibroadenomas and cancer.
\nScatter plot showing the distribution of cancers (squares), Fibroadenomas (diamonds), cysts (triangles) and other benign (circles).
Cyst, fibroadenoma, cancer: Waveform SS images showing well circumscribed margins and smooth internal textures for both the 1.5 cm cyst in dense white breast tissue (left) and the 0.7 cm fibroadenoma (middle) in darker fat. The 1.8 cm cancer (right) has irregular margins, heterogeneous content and subtle peritumoral spiculations.
In this study we reviewed the status of breast cancer screening and the potential role that ultrasound tomography (UST) could play in breast imaging. Several results from recent ongoing UST studies were used in this review. The main conclusions from those studies are:
UST sound speed demonstrated a high degree of correlation of breast tissue structures relative to fat subtracted contrast-enhanced MRI. This correlation of structures was most evident in the coronal plane comparisons.
UST can scan ~90% of the volume of the breast compared to MRI. With proper positioning UST can image the pectoralis muscle and a portion of the axillary tissue.
UST demonstrated a spatial resolution of 0.7mm in the coronal plane, similar to MRI.
Initial clinical results suggest an ability to characterize lesions using margin boundary scores in combination with sound speed and attenuation parameters. These parameters leverage all three imaging modes of UST (reflection, sound speed and attenuation).
UST is a promising new modality that has the potential to complement existing breast imaging methods to aid in lesion detection and characterization. Future larger scale studies will assess UST’s role in diagnostic and screening settings.
\nThe authors thank Dr. Mark Sak for providing images relating to the MRI comparison study and Mr. Mark Krycia for his help in the statistical analysis of the lesion characterization data. The work presented in this paper was supported by NIH grant 5R44CA165320-05.
\nSediment transport is the movement of solid particles driven by fluid like water or wind in rivers, lakes, reservoirs, coastal waters. Generally, in the real world the flow is unsteady like flood waves, tidal waves and wind waves, because steady and uniform flows are very rare in reality. Even so, it is understandable that sediment transport is first observed under well controlled conditions in laboratory, and then the data are collected to calibrate the models. These formulae are further examined using field data by assuming the laboratory flow conditions (generally steady and uniform flows) can be extended to rivers and coastal waters (generally unsteady and non-uniform).
In the literature, many formulae use the boundary shear stress
where gb and gt = bed-load and total load of sediment discharge per unit width, g = gravitational acceleration, d50 = median sediment size, ρs = sediment density, and ρ = water density, h = water depth, S = energy slope, cf = friction factor which is constant in fully rough regime. The subscribes b and t denote the bed load and the total load. Eqs. (1) and (2) demonstrate that if d50, ρs are constant, sediment discharge only depends on τ.
Alternatively, the mean velocity U was selected to represent the hydraulic parameter for sediment discharge or concentration like the Velikanov’s [6] parameter, U3/(ghω). The WIHEE’s [7] equation which has been widely used in China has the following form:
where C = sediment concentration, k1 and m are empirical coefficient, ω = sediment settling velocity.
Besides the parameters U and τ alone, attempts have been made to correlate the sediment transport with the product of U and τ. Probably Bagnold [8] was the first one to do so, and it is known as the stream power (= τU). Likewise, the product of U and S, or the unit stream power US/ω was used by Yang [9]. van Rijn [10] selected u*’, the shear velocity related grains, in his equations, i.e., T, and d*, they are
where
where the critical shear stress τc = ρu*c2, ν = kinematic viscosity.
Yang and Tan [11] found that the shear velocity u*’ is responsible for transporting the sediment particles, Yang [12] defined the energy dissipation on sediment transport as E = τu*’, and obtained the formula of sediment transport:
where the arrows represent the direction, i.e., sediment is transported in the same direction as the near bed flow if the flow directions of upper and lower layers are different, Ec (= ρu*c3), k is a constant (= 12.2) and insensitive to other hydraulic parameters like Froude number, Reynolds number, relative roughness and Rouse number [13].
Obviously, the hypothesis in all equations listed above is that the higher the streamwise parameters are (e.g., U, u*’, τ, E or US etc.), the more particles are transported [14]. However, this prediction is invalid in unsteady conditions [15, 16]. Tabarestani and Zarrati [17] reviewed the performance of existing formulae and concluded that in general, the sediment discharge under unsteady flow conditions cannot be predicted by these equations, because the streamwise parameters in the rising limb is much larger than that in the falling limb, but the measured sediment load yield during hydrograph rising limb is smaller than that in the falling limb. The highest gt or C comes after the peak flowrate or velocity U, and the lag phenomenon has been widely observed and reported. The shear stress based theory has also been questioned by Nelson et al. [16] who observed from their experiment that the sediment flux increases even though the bed shear stress decreases.
Sleath [18] argued that when the “pressure gradient” is not small compared with the shear stress exerted by the flow, these equations need to be modified and a new
where σ is the angular frequency of waves.
Alternatively Francalanci et al. [19] suggest using the pressure P to express the unsteadiness, but Liu and Chiew [14] and Cheng and Chiew [20] use the hydraulic gradient i in the sediment layer. The challenge also comes from the bursting phenomenon even in steady and uniform flows. It is found that the similar lag phenomenon exists in a bursting cycle [21, 22]. Cellino and Lemmin’s [23] experiments demonstrate that the upward flow (or ejection) appears responsible for the threshold of particle movement, the entrainment and transport of bedload and lifting of sediment into suspension. This cannot be explained by the parameters of pressure P or hydraulic gradient i or seepage velocity.
It seems that there is a knowledge gap between the unsteady flows and sediment transport, a new parameter is needed to be developed to express the unsteadiness, thus the above phenomena can be explained. In this study, the induced vertical velocity V is selected to express the effect of unsteadiness on the sediment, an attempt is made to justify its suitability for sediment transport as well as the phenomena of phase-lag and bedform formation. The research objectives include:
to compare V with other parameters to express the force induced by unsteady flows;
to establish a simple connection between Vb in the sediment layer and V in the main flow;
to develop formulae to express critical shear stress, sediment discharge and Rouse number in unsteady flows;
to explain the mechanism of phase lags and bedform formation.
The chapter discusses the existence of vertical velocity in unsteady flows first, then the influence of vertical velocity on critical shear stress of sediment is analyzed, followed by its influence on sediment discharge and suspension concentration. Finally a comprehensive discussion is provided.
Sediment transport is a joint result of streamwise and vertical motions of fluid. This joint effect can be seen from the definition of Shields number that is the ratio of forces in streamwise and vertical directions as noted by Francalanci et al. [19]:
where τ* = Shields number. The numerator denotes the streamwise friction force and the denominator represents the vertical force, i.e., the net buoyant force of particle. Sediment starts to move at τ* ≥ τ*c, the critical Shields number.
A simple wave model is shown in Figure 1a where a surface wave induces a vertical motion for the particles on the permeable bed. The surface wave is propagating in the research domain where the current velocity is U, the streamwise parameters like the point velocity, shear stress, pressure P and hydraulic gradient i in the soil are also modified. In this study, the induced vertical motion has been expressed by velocity at the interface is Vb. The relationship between the wave and its induced vertical velocity is shown in Figure 1b.
(a) Schematic diagrams showing interaction of surface waves and induced and vertical motions at the sediment layer along x direction. (b) Definition of progressive wave and its induced vertical velocity at different time (x = constant).
In Figure 1, the continuity equation of unsteady flows must be satisfied, i.e.,
where u and v are the streamwise and vertical time-averaged velocities in x and y directions. The vertical velocity can be determined from Eq. (10) as follows:
In Eq. (11) the term ∂u/∂x is the gradient of streamwise velocity in x-direction, it is positive if the velocity becomes higher to downstream (accelerating), and negative if the fluid particles experience decelerating. Hence, the accelerating flow yields a negative or downward v, the decelerating flow generates an upward or positive v.
At the permeable boundary, the fluid velocity must meet the continuous boundary condition, i.e., v(y=0+) = v(y=0−), or the velocity inside the sediment layer must be same as the velocity in the main flow at the interface. Thus it can be concluded that a downward velocity exists in the sediment layer when the main flow layer is accelerating, and an upward velocity appears when a flow is decelerating.
Generally speaking, the rising limb is the accelerating stage which induces a downward velocity, but the decelerating stage in ebb limb generates an upward velocity. In the real world, it is also possible that flows in both rising/falling limbs are accelerated as observed by Song and Graf [24], who used acoustic Doppler velocity profiles measured the vertical velocity in unsteady open channel flows, and found during the rising/falling limbs, “the measured vertical velocity are almost always negative, and this implies that the flows of the present experiments are accelerating ones”. On the other hand, Leng and Chanson [25] used an acoustic Doppler velocimeter (ADV) measured the vertical velocity in tidal bores and found that the vertical velocity is always upward or decelerating in both rising and falling limbs. To simplify the discussion, this study only discusses the cases shown in Figure 1b and the waves’ influence on parameters like q, U is assumed to be negligible.
The direction of vertical velocity can noticeably change the profile of Reynolds shear stress, streamwise velocity etc. [12, 26]. One of the examples is shown in Figure 2, Kemp and Simons [27, 28] measured the velocity profiles in a flume where the incident wave was set to propagate against or along the direction of the currents. The flow depth at the test section was kept at 200 mm for all tests. Regular waves were generated with a constant wave period of 1 second. The wave heights were 27.9 to 20.7 mm, the wave lengths were 1053 mm to 1426, respectively. Their results clearly show that the measured velocity is greater than log-law’s prediction when waves opposite the current as the original uniform flow is decelerated by the waves from downstream, but less than the log-law’s prediction when the waves to the currents as the original uniform flow is accelerated by waves from upstream. Existing research [26, 29] shows that in a turbulent flow the log-law is satisfied if the upward velocity in the main flow V = 0, but the measured velocity is higher than the log-law’s prediction if V > 0 or upward velocity exists, and the maximum velocity is submerged if V < 0 (or downward velocity exists). Further investigation shows that a decelerating flow generates an upward velocity, but an accelerating flow induces downward velocity [30]. Therefore one can infer that in Figure 2, there exists an upward velocity for waves against a current or the waves make the current decelerated; but a downward velocity exists in the case of waves following current, which accelerates the water.
Deviation of measured velocity from log-law by Kemp and Simons [21, 22].
For sediment particles in Figure 1, the settling velocity ω in still water is determined by:
where drag coefficient Cd depends on the Reynolds number Re (= ωd/ν) and Cd = 0.45 for large Reynolds number, i.e., Re >1000.
If a surface wave induces an upward velocity Vs in the preamble sediment layer, the net settling velocity is reduced to ω - Vb. The reduction of settling velocity could be treated by altering its density from ρs to ρs’ by assuming the particle’s size remains unchanged, and the force balance equation is similar to Eq. (12) with the following form:
From Eqs. (12) and (13), one can derive the following relationship:
where α = Cd′/Cd and α = 1 are assumed to simplify the mathematical treatment. Eq. (14) shows that if Vb is upward, then ρs’ < ρs, or the sediment particles become lighter in the “boiling” environment. If the upward Vb = ω, Eq. (14) shows that the sediment density is similar to the water density ρs’ = ρ. If the sediment particles are exercising the downward velocity (negative Vb), then the density ρs’ > ρs, or the sediment behaves like heavy metals. As the decelerating velocity can generate upward velocity, it can be inferred that if the streamwise parameters keep almost unchanged, the sediment can be more easily transported in decelerating flows relative to the accelerating flows. In other words, the sediment particles become lighter in decelerating flows (or decelerating phase), but heavier in accelerating flows/phase. As Eq. (10) is also valid for turbulent velocity and wave conditions, then the conclusion can be extended to the bursting phenomenon or wave conditions where the accelerating/decelerating phases alternate randomly or regularly, thus these equations provide a general tool to analyze sediment transport.
It is interesting to discuss how the waves affect the initiation of sediment movement. For an unsteady flow, the existing Shields diagram may be invalid to express the threshold sediment motion, due to the existence of vertical velocity caused by its unsteadiness. When the apparent sediment density is included in the Shields number, it has the following form:
where τc’ is the critical shear stress with vertical velocity. Inserting Eq. (14) into Eq. (15), one has:
Using Eq. (9), Eq. (16) can be rewritten as follows
Eqs. (16) and (17) generally express the relationship between the Shields number τ*’ with waves and the original Shields number τ* without waves. It predicts that the original Shields number may significantly deviate from the Shields curve subject to wave conditions.
Eq. (15) includes the influence of the vertical velocity, it demonstrates that the upward velocity reduces particles’ apparent density, thus the required critical shear stress will be also reduced. Whilst the downward velocity increases the apparent density, thus the required critical shear stress is higher. If the cases with/without vertical velocity are compared, the critical shear stress without waves τc and the critical shear stress with waves τc’ have the following relationship:
and
Eq. (18) shows that the critical shear stress τc’ in unsteady flows. It should be stressed that for sediment incipient motion, Vb in Eq. (19) depends on the instantaneous maximum upward velocity for which the ejection of burst phenomenon, unsteadiness and others may jointly contribute. For flows shown in Figure 2, one can infer that the measured τc’ is less than Shields diagram’s prediction when the waves propagate against the current, but the τc’ becomes larger than τc when the waves propagate with the current. The reason is that, the former generates an upward velocity in the decelerating flows, but the latter has a downward as it is an accelerating flow.
If the influence of small wave on the shear stress is negligible, the Y with small waves must be higher than the Y without waves. In such case, one can easily conclude from Eq. (18) that the τc’ (with waves) must be always less than τc (without waves). In the literature, it seems that many researchers agree that the existing Shields diagram can be extended to the wave-current motion (i.e., [31, 32]). Till recently, few researchers like Green and MacDonald [33] found waves, not currents initiate sediment transport, their data show that “observed τ*’ never exceeded the theoretical dimensionless τ*”. It is well known that for large particles, the critical Shields number τ* = 0.06. They observed suspension at the same value of τ* when waves are present in tidal flows, similar observations were reported by Green and Coco [34]. All of these observations can be easily explained by Eq. (18) when Y ≈ 1.
It should be stressed that accelerating flows constrain sediment mobility from vertical point of view, but the higher velocity and shear stress in the rising limb promote sediment transport in the streamwise direction, therefore the complete effect of accelerating flows in the rising limb should include both shear stress and maximum Y. Likewise, the decelerating flow makes particles “lighter” in vertical direction, but the reduced shear stress makes particles to move “harder”. Therefore, one need to justify the critical shear stress by considering both streamwise and vertical parameters.
Eq. (18) clearly demonstrates that the critical shear stress is jointly determined by the streamwise and vertical motions. The coexistence of streamwise/vertical motions results in the invalidity of Shields diagram which can be improved by Eq. (18) and shown in Figure 3, where the Shields number in the original Shields diagram is τ* = 0.045, 0.03 and 0.13 are calculated using Eq. (18). The region below the curves represents that the sediment is static, and above these curves is mobile. The calculated results show that when Y ≥ 0.7, the sediment is mobile, for which the required shear stress is always zero.
Sediment incipient conditions in wave conditions, the required shields numbers depend on the veritocal motion, i.e., ±Y. based on Eq. (15), the calculated solid line (——) represents non-cohensive sediment in shields diagram τ*’ = 0.045; the dotted line (⋯⋯⋯) for τ*’ = 0.03; and the dashed line (– –) for very fine sediment with τ*’ = 0.13. Below these curves, particles remain static, above the curves particles are in mobile state.
As mentioned before that sediment transport is a joint effect of streamwise and vertical motions, the latter can be represented by the apparent sediment density. Therefore, Eq. (7) can be modified with the following way:
For sediment transport in waves conditions, the bed shear stress τ = τw + τcu and near bed velocity ub = uw + ucu, where the subscripts w and cu refers to waves and currents. Yang [12] obtained the formula which agrees reasonably well with von Rijn’s data in 1993, 1995 and 1999 for sediment transport when waves follow or oppose the currents, there are some angles between the direction of wave propagation and current, and waves are broken over a near shore bar, respectively. Even the best agreement has been achieved among the existing formulae, noticeable discrepancies imply that some mechanism of sediment transport by waves needs further investigations.
Eq. (20) shows that the direction of sediment motion is always the same as the near bed velocity. This is meaningful to specify the sediment moving direction in coastal waters where the direction of flow in up layer is often different from that in the bottom layer. Eq. (20) has the following simplified form [35]:
Inserting Eq. (12) into Eq. (21), one has:
Eq. (22) shows that sediment transport rate is jointly determined by the streamwise flow conditions (i.e., τo and u*’) and Y.
For the maximum over-the-wave-cycle horizontal wave-orbital speed at the bed Ub can be expressed by the wave height H and the wave period T, as both these govern the wave-orbital speed at the bed at any given water depth h. For linear waves, this is expressed as
where the dispersion relationship gives:
σ = 2π/T, ko is the wave number and ko = 2π/L, L = wave length.
It can be assumed that at the interfacial boundary, v(y = 0+) has the same magnitude order as Ub, and the vertical velocity at the sediment layer can be expressed as
To evaluate the influence of vertical velocity on sediment transport rate, one can compare the sediment transport rate in two cases: with or without the vertical velocity induced by waves if τo remain unchanged. At Vb = 0, Eq. (22) becomes:
From Eqs. (22) and (26), one has:
where gt = Cq and C = sediment concentration. For a current with very small waves, the influence of small waves on the discharge q is negligible, thus gt(Y)/gt(0) ≈ C(Y)/C(0).
Green [36] measured sediment concentration in an estuarine intertidal flat in New Zealand under very small waves. The wave height is less than 10 cm, and wave period ranges from 1.0–1.8 s. The measured data shows that sediment concentration in the rising tide is not very high, the highest concentration is always appear in the ebb tide. Eq. (11) may provide an explanation when the rising tide is assumed to be accelerating and the falling tide is decelerating. A downward velocity is generated the rising tide, which has the same effect on sediment as the particle’s density becomes heavier. But during the falling limb or low tide, the particles become lighter, so the concentration becomes higher in this stage as shown in Figure 4.
Measured sediment concentration normalized by C(0) = 5 mg/L versus the wave-orbital acceleration normalized by 23 cm/s2. The raw data were deprived from Green [17], the acceleration in flood limb was set to negative and the acceleration in ebb stage was set to be positive. After this transformation, the obtained data can structurally match Eq. (27), implying the connection between the dimensionless parameters Y and the wave-orbital acceleration.
In their analysis, Green [36] found that the “wave-plus-current-stress” theory provides poor agreement with their data. But the “wave-orbital speed” theory performed the best at predicting the incipient motion and suspension. They found a strong relationship between the measured sediment concentration and the wave-orbital acceleration a0 which is defined as:
where n = the number of zero-down crossing waves in the burst, UZDC+ is the maximum zero-down crossing current excursion in the positive direction from its average velocity. UZDC- is the maximum zero-down crossing current excursion from the mean velocity in the negative direction, TZDC is the period for the events.
Figure 4 shows a plot C(Y)/C(0) versus Y (=a0/23). Green [36] plotted his measured concentration in mg/L against a0 using Eq. (28), in which the wave period is almost constant, thus the acceleration a0 in Eq. (28) is actually the velocity. In Figure 4, the averaged concentration in the flood stage is used as C(0) and C(0) = 5 mg/L. It is found that data points match Eq. (27) very well when the acceleration a0 is normalized by 23 cm/s2 that is not clear the reason. In the calculation, the sediment and seawater densities are 2650 and 1025 kg/m3, respectively.
It can be seen that the sediment transport rate can be significantly promoted by an ebb tide, if the upward velocity is 75% of settling velocity (Y = 0.75), then the predicted sediment transport rate can be increased to 27 times of gt(0). Figure 4 also shows that the sediment transport rate is slightly reduced if a downward flow exists. If Y = −0.5, then the sediment transport rate will be reduced to 1/2 of gt(0), this transport rate is achieved as the particles becomes “heavier”.
The governing equation of suspended concentration can be derived from the continuity equation of solid-phase in the following form [37].
where c’ = fluctuation of sediment concentration; c = local time-averaged sediment concentration, u, v and w are the streamwise, vertical and lateral time-averaged velocities; u’, v’ and w’ are the velocity fluctuations in y and z directions, respectively.
In equilibrium conditions, time averaging of Eq. (29) gives:
The integration of Eq. (30) with respect to y yields the following equation
If the eddy viscosity is used and Rouse number in Rouse’s law has the following form:
Similar to the Shields number, many researchers also found that the measured Z is different from the calculated Z. van Rijn [10] and Van de Graaff [38] attribute this invalidity to sediment characteristics like size or streamwise flow strength, Eq. (33) indicates that if the vertical velocity exists, it also leads to the invalidity of Rouse number in practice.
Rosea and Thorneb [39] observed the Rouse number by measuring suspended sediment concentration profiles in the river Taw estuary, UK, where the flow is dominated by strong rectilinear, turbulent tidal currents. Their measurement was focused on the rising (flood) tide for a period of 3 hours. The measured Z(Y)/Z(0) is shown in Figure 5, at the at the starting point the minimum vertical velocity Y can be expected, and Z(Y)/Z(0) ≈ 1 is observed, in the process, the streamwise velocity or shear velocity changed in a range of ±20%, but the observed Z(Y)/Z(0) increased 150% and all data points shows that Z(Y)/Z(0) >1 in the rising tidal flow, this is agreed with Eq. (33), i.e., accelerating flows generate an downward velocity or negative Y that constrains sediment transport. This also can be seen from the measured sediment concentration Ca at the reference level near the sea bed, the decreasing Ca implies that the downward velocity makes the particles “heavier” to move, consequently Ca is reduced to 44.8% of its original value.
Measured Z(Y)/Z(0) in a rising tidal flow by Rose and Thome, at the starting point the streamwise velocity was the highest, minimum vertical velocity Y can be inferred, and Z(Y)/Z(0) ≈ 1 is observed and all data points show Z(Y)/Z(0) >1 in the rising tidal flow.
If Z(Y)/Z(0) in flood tide is compared with its values observed during ebb-tidal, Eq. (33) clearly indicates that the ebb-tide will have a lower value. This is in agreement with Al-Ragum’s [40] observation as shown in Table 1. The data were collected from the Biscay Bay near Spain and France border. “The Rouse parameter varied with the tide, and the values were higher on the flood-tide than on the ebb-tide” as claimed by the author. The average Rouse parameter during flood tide is about 0.7, but it is reduced to 0.44 during the ebb tide. The flood tide generates 60% higher Rouse number relative to that during the ebb-tide.
Profile | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 1 | 2 | 3 | 5 |
---|---|---|---|---|---|---|---|---|---|---|---|---|
Tide | ebb | Flood | ||||||||||
Rouse number | 0.55 | 0.45 | 0.45 | 0.50 | 0.27 | 0.35 | 0.48 | 0.50 | 0.68 | 0.95 | 0.7 | 1.2 |
Average | 0.44 | 0.70 |
Measured rouse numbers (ω/κu*) in flood-tide and ebb tide by AL-Ragum [3].
For sediment transport by either flood waves in rivers or tidal waves in the sea, the unsteadiness plays a significantly role for sediment transport. The equations developed from steady flow may be invalid in unsteady flows. Some researchers like Graf and Suszka [41] found that the measured sediment transport rate in an unsteady flow is always larger than these equations’ predictions. An unsteadiness parameter was proposed by them:
where td is the duration of a hydrograph, h1 is the initial or baseflow depth, hp is the peak flow depth of the hydrograph.
It is interesting to note that (hp-h1)/td is actually the averaged vertical velocity V. Eq. (34) can be understood as the ratio of vertical velocity to the shear velocity, similar to Y in Eq. (19). The unsteadiness parameter P1 is useful for the prediction of time average sediment transport rate, but it cannot be used to explain the measured instantaneous rate gt or concentration C that depends on the instantaneous vertical velocity, thus Eq. (19) may have a wider application. Compared with Eq. (34), Eq. (19) is simple and direct, the difficult parameter u* is replaced with the sediment settling velocity ω that is independent of flow characteristics, and the instantaneous vertical velocity V can easily explain the observed phenomena in unsteady flows.
In fact, the unsteadiness parameter S1 in Eq. (34) can be written in its alternative form:
where a is the wave amplitude similar to hp-h1 in Eq. (34) and the vertical velocity V = aσ/(2π), thus Eq. (35) shows that S1 is similar to Y. Sleath [18] also proposed another parameter to express sediment transport by waves, i.e., ω/σδ, and δ is the maximum thickness of the mobile layer, which can be read as 1/Y if Vb = σδ is assumed.
Figures 4 and 5 show the influence of unsteadiness on sediment transport in tidal flows. For flood waves in a river, Lee et al. [15] measured the transport rate over a series of triangular hydrographs. Their experimental results show the existence of phase lag between the peak discharge and peak sediment rate gt, which lag is very long and about 6–15% of the flow hydrograph duration. Figure 6 shows the hydrograph and measured sediment discharged by Lee et al. [15], it shows that the highest sediment transport rate appears in the falling stage when the shear stress is much less than that in the rising stage. This phase-lag phenomenon cannot be explained by those shown in Eqs. (1)–(7). It is interesting to note that there are two gt peaks in Figure 6, the mechanisms may be totally different, the former in the rising limb is likely generated by very high τ in the rising limb, but the upward velocity probably dominates the second peak where the shear stress is very small.
Sediment transport rate and flood hydrograph measured by Lee et al. [23].
It should be mentioned that the peak sediment discharge in the rising limb is not always discernible as shown in Figure 6. For example, Qi et al. [42] reported that in Yellow River, artificial flood waves have been used to flush sediment in lower course of Yellow River by releasing water from its Xiaolangdi reservoir. As shown in Figure 7, the rising limb did not increase sediment concentration much, but the falling stage generated very high sediment concentration. From their experience, to enhance the flush efficiency, the duration of rising limb should be short as its rising flow does not increase gt or C too much.
Measured sediment concentration over a hydrograph at Huayuankou, Yellow River from July 4–6, 2010 by Qi et al. [31].
To interpret the results in Figures 6 and 7, the conceptual mathematical model in Eq. (27) may be useful as it covers the parameters in streamwise and vertical directions. Eq. (27) precisely suggests that the upward velocity may be responsible for the widely observed “phase lag” in sediment transport in rivers.
The formation of ripples and dunes over a flat mobile bed is an amazing phenomenon, and has attracted many investigations. All previous equations of sediment transport (e.g., Eqs. (1)–(7) and Eqs. (34) and (35) fail to explain how the bedforms are formed, because these equations only use the streamwise parameters (U, τ etc.) that are constant in every cross section from upstream to downstream if the flow is steady and uniform, thus the sediment discharge in every cross section is same and no local erosion occurs, so none of them can successfully explain the formation of ripples and dunes.
However, Eqs. (14) and (22) may provide a possible explanation for the discontinuity of sediment transport from upstream to downstream. It is well known that turbulence in a steady and uniform flow is dominated by complex, multiscaled, quasi-random and organized eddies that possess both spatial and temporal coherence [43]. The velocity fluctuations are also governed by the continuity equation with the following form:
The coherent events can be broadly divided into ejections (v’ > 0 or decelerating) and sweeps (v’ < 0 or accelerating), both of them are always alternated in space and time.
To help conceive the formation of bedforms, a flow region in Figure 8 is divided simply into three zones, A, B and C during a short period. If the flow region B is dominated by the ejection event (denoted by “+” in Figure 8 for upward vertical velocity), severe erosion should be observable in Zone B as Eq. (27) and Figure 4 indicate that the upward velocity significantly promotes the sediment discharge. On the other hand, Zones A and C are dominated by the downward velocity (or negative “-” velocity), and Eq. (27) and Figure 4 predict that the sediment carrying capacity is weaker if the vertical velocity is negative, therefore the sediment from zone B has to deposit at Zone C. It can be seen that the vertical velocity and its alternation in direction in space play a key role for the formation of dunes and ripples. The discontinuity of sediment-laden capacity along the flow direction is uneven, this triggers the formation of bedforms, once some scouring holes are formed over a flat mobile bed, erosion in these areas most likely would continue till the equilibrium condition is reached.
Relationship between the alternative vertical velocity and bedform formation, where “+” sign denotes upward velocity in region B and “-” is the downward velocity in region A and C. The dotted vertical lines denote the flow region division lines, the open circles denote the sediment particles, the solid circles denote that particles’ density “becomes heavier”, and dashed circles denote the “lightweight sediment”, the open circles are normal sediment particles.
Alternatively, we can consider a simple model that all particles in Figure 8 possess higher apparent density in zone A and C like iron particles (represented by dark solid circles in Figure 8), but the particles in zone B have lighter density (like plastic particles). All particles in zone A, B and C have the same diameter. It is understandable that a scour hole will be formed in zone B, and deposition will occur at C even though the U and τ remains constant in zones A to C. In other words, it can be seen that the vertical velocity and its spatial alternation play a key role for the formation of bedform. The simple model shown in Figure 8 explains the formation of a scour hole on a flat plane that triggers the formation of bedforms. This mechanism can be extended to dune formation in deserts where the horizontal wind generates sediment transport in horizontal direct, and vertical motions yields the bedforms. The wind is accelerating along the upwind side of a dune, thus its surface is smooth, and the decelerating wind after the peak generates upward velocity, thus small holes are formed in the lee side.
By comparing the mechanism of phase lag and bedform formation discussed above, one may find that the vertical velocity is responsible for both phase lag phenomenon and bedform formation. The temporal alternation of upward and downward velocities generates the phase lag phenomenon, whilst its spatial alternation yields the bedforms. Generally speaking, we can see that the phenomena of sediment transport can be categorized into streamwise and vertical motions dominated events. Sediment transport should be expressed using variables in streamwise and vertical directions jointly.
Generally, all interfaces on solid–liquid, liquid–liquid, liquid-gaseous phases exist waves if there exists alterative vertical motions as shown in Figure 8, otherwise no waves can be observed no matter how high the velocity is if the flow is laminar. Likewise, the ocean waves between water and air are not caused by the shear stress or wind velocity on the sea surface, but the air pressure oscillation whose period should be identical to the ocean waves. In other words, turbulence is the cause of ocean waves. In summer, the heated sea surface generates an upward motion, consequently typhoons, cyclones and hurricanes can be observed. In winter, the downward cold air yields a relatively calm surface.
The existence of upward velocity can be inferred from numerous small bubbles when waves are broken. The soluble gas or air near in a lower lever like the seabed (high pressure) can be transferred to the surface (gauge pressure = 0) by the upward velocity, which causes significant pressure difference of inside and outside bubbles, consequently the bubbles are broken. In other words, from bubbles one may conclude that there is an upward velocity to transfer the bubbles from deep water to the surface, this is also true for bubbles in hydraulic jumps. It is predictable that in high speed flow, cavitation (i.e., local scour over a metal/concrete surface) can be observed when decelerating flow or the vertical flow exists. The liquefaction can be observed when the seepage velocity and particle settling velocity are in the same order of magnitude.
This study investigates the influence of vertical velocity induced unsteady flows on sediment transport. It is well-known that the vertical velocity is ubiquitous and it can be induced by coherent structures, non-uniformity, unsteadiness, and so on. This paper just discusses the simplest cases, i.e., the presence of vertical velocity does not significantly alter the streamwise parameters like velocity U or discharge q, in which the rising limb or accelerating flow generates a downward velocity, but the falling limb or decelerating flow induces an upward velocity. A conceptual mathematical model is developed to account for the vertical velocity’s influence on particles’ critical shear stress, sediment discharge and suspension. It is found that the model can provide a qualitatively explanation to some observed phenomena. Based on this investigation, the following conclusions can be drawn:
The upward velocity enhances sediment mobility and downward velocity increases its stability. Mathematically the behavior of sediment transport subject to a vertical motion can be equivalently treated by the variation of apparent density. Particles become “heavier” when they experience the downward flows, this reduces the sediment transport rate. But particles become “lighter” in flows with upward velocity where the sediment discharge is increased significantly. The obtained new equation for sediment transport’s apparent density is used to explain sediment transport in unsteady flows.
The application of Shields diagram, equations of sediment discharge and Rouse equation developed from steady flows could be extended to unsteady flows if the vertical parameter Y (= V/ω) is included. The conceptual model shows that sediment is easily be transported when Y > 0, but difficult to move when Y < 0, same for the transport rate gt and Rouse number Z. The developed equations provide reasonably good agreement with the measured data. The condition for liquefaction can be expressed by Y = 1.
The mathematical model may also provide a tool to understand many odd phenomena in sediment transport like the phase lag phenomenon and bedform formation. Both are widely reported and discussed, this is the first trail to give the similarities between these two phenomena. The research shows that the temporal variation of vertical velocity results in the phase lag, and its spatial variation leads to the bedform formation.
In the literature, the vertical velocity is generally ignored in the measurement, which leads to that the conclusions listed above rest on the inferences of vertical velocity, not its measured values and direction. In future, systematical experiments are needed to investigate its role in order to validate the conceptual model.
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\\n\\n5. TERMINATION
\\n\\n5.1 IntechOpen has a right to terminate this Publication Agreement for quality, program, technical or other reasons with immediate effect, including without limitation (i) if the Corresponding Author or any Co-Author commits a material breach of this Publication Agreement; (ii) if the Corresponding Author or any Co-Author (being an individual) is the subject of a bankruptcy petition, application or order; or (iii) if the Corresponding Author or any Co-Author (being a company) commences negotiations with all or any class of its creditors with a view to rescheduling any of its debts, or makes a proposal for or enters into any compromise or arrangement with any of its creditors.
\\n\\nIn case of termination, IntechOpen will notify the Corresponding Author, in writing, of the decision.
\\n\\n6. INTECHOPEN’S DUTIES AND RIGHTS
\\n\\n6.1 Unless prevented from doing so by events outside its reasonable control, IntechOpen, in its discretion, agrees to publish the Chapter attributing it to the Corresponding Author and any Co-Author.
\\n\\n6.2 IntechOpen has the right to use the Corresponding Author’s and any Co-Author’s names and likeness in connection with scientific dissemination, retrieval, archiving, web hosting and promotion and marketing of the Chapter and has the right to contact the Corresponding Author and any Co-Author until the Chapter is publicly available on any platform owned and/or operated by IntechOpen.
\\n\\n6.3 IntechOpen is granted the authority to enforce the rights from this Publication Agreement, on behalf of the Corresponding Author and any Co-Author, against third parties (for example in cases of plagiarism or copyright infringements). In respect of any such infringement or suspected infringement of the copyright in the Chapter, IntechOpen shall have absolute discretion in addressing any such infringement which is likely to affect IntechOpen's rights under this Publication Agreement, including issuing and conducting proceedings against the suspected infringer.
\\n\\n7. MISCELLANEOUS
\\n\\n7.1 Further Assurance: The Corresponding Author shall and will ensure that any relevant third party (including any Co-Author) shall, execute and deliver whatever further documents or deeds and perform such acts as IntechOpen reasonably requires from time to time for the purpose of giving IntechOpen the full benefit of the provisions of this Publication Agreement.
\\n\\n7.2 Third Party Rights: A person who is not a party to this Publication Agreement may not enforce any of its provisions under the Contracts (Rights of Third Parties) Act 1999.
\\n\\n7.3 Entire Agreement: This Publication Agreement constitutes the entire agreement between the parties in relation to its subject matter. It replaces and extinguishes all prior agreements, draft agreements, arrangements, collateral warranties, collateral contracts, statements, assurances, representations and undertakings of any nature made by or on behalf of the parties, whether oral or written, in relation to that subject matter. Each party acknowledges that in entering into this Publication Agreement it has not relied upon any oral or written statements, collateral or other warranties, assurances, representations or undertakings which were made by or on behalf of the other party in relation to the subject matter of this Publication Agreement at any time before its signature (together "Pre-Contractual Statements"), other than those which are set out in this Publication Agreement. Each party hereby waives all rights and remedies which might otherwise be available to it in relation to such Pre-Contractual Statements. Nothing in this clause shall exclude or restrict the liability of either party arising out of its pre-contract fraudulent misrepresentation or fraudulent concealment.
\\n\\n7.4 Waiver: No failure or delay by a party to exercise any right or remedy provided under this Publication Agreement or by law shall constitute a waiver of that or any other right or remedy, nor shall it preclude or restrict the further exercise of that or any other right or remedy. No single or partial exercise of such right or remedy shall preclude or restrict the further exercise of that or any other right or remedy.
\\n\\n7.5 Variation: No variation of this Publication Agreement shall be effective unless it is in writing and signed by the parties (or their duly authorized representatives).
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\\n\\nAny modification to or deletion of a provision or part-provision under this clause shall not affect the validity and enforceability of the rest of this Publication Agreement.
\\n\\n7.7 No partnership: Nothing in this Publication Agreement is intended to, or shall be deemed to, establish or create any partnership or joint venture or the relationship of principal and agent or employer and employee between IntechOpen and the Corresponding Author or any Co-Author, nor authorize any party to make or enter into any commitments for or on behalf of any other party.
\\n\\n7.8 Governing law: This Publication Agreement and any dispute or claim (including non-contractual disputes or claims) arising out of or in connection with it or its subject matter or formation shall be governed by and construed in accordance with the law of England and Wales. The parties submit to the exclusive jurisdiction of the English courts to settle any dispute or claim arising out of or in connection with this Publication Agreement (including any non-contractual disputes or claims).
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The Corresponding Author (acting on behalf of all Authors) and INTECHOPEN LIMITED, incorporated and registered in England and Wales with company number 11086078 and a registered office at 5 Princes Gate Court, London, United Kingdom, SW7 2QJ conclude the following Agreement regarding the publication of a Book Chapter:
\n\n1. DEFINITIONS
\n\nCorresponding Author: The Author of the Chapter who serves as a Signatory to this Agreement. The Corresponding Author acts on behalf of any other Co-Author.
\n\nCo-Author: All other Authors of the Chapter besides the Corresponding Author.
\n\nIntechOpen: IntechOpen Ltd., the Publisher of the Book.
\n\nBook: The publication as a collection of chapters compiled by IntechOpen including the Chapter. Chapter: The original literary work created by Corresponding Author and any Co-Author that is the subject of this Agreement.
\n\n2. CORRESPONDING AUTHOR'S GRANT OF RIGHTS
\n\n2.1 Subject to the following Article, the Corresponding Author grants and shall ensure that each Co-Author grants, to IntechOpen, during the full term of copyright and any extensions or renewals of that term the following:
\n\nThe aforementioned licenses shall survive the expiry or termination of this Agreement for any reason.
\n\n2.2 The Corresponding Author (on their own behalf and on behalf of any Co-Author) reserves the following rights to the Chapter but agrees not to exercise them in such a way as to adversely affect IntechOpen's ability to utilize the full benefit of this Publication Agreement: (i) reprographic rights worldwide, other than those which subsist in the typographical arrangement of the Chapter as published by IntechOpen; and (ii) public lending rights arising under the Public Lending Right Act 1979, as amended from time to time, and any similar rights arising in any part of the world.
\n\nThe Corresponding Author confirms that they (and any Co-Author) are and will remain a member of any applicable licensing and collecting society and any successor to that body responsible for administering royalties for the reprographic reproduction of copyright works.
\n\nSubject to the license granted above, copyright in the Chapter and all versions of it created during IntechOpen's editing process (including the published version) is retained by the Corresponding Author and any Co-Author.
\n\nSubject to the license granted above, the Corresponding Author and any Co-Author retains patent, trademark and other intellectual property rights to the Chapter.
\n\n2.3 All rights granted to IntechOpen in this Article are assignable, sublicensable or otherwise transferrable to third parties without the Corresponding Author's or any Co-Author’s specific approval.
\n\n2.4 The Corresponding Author (on their own behalf and on behalf of each Co-Author) will not assert any rights under the Copyright, Designs and Patents Act 1988 to object to derogatory treatment of the Chapter as a consequence of IntechOpen's changes to the Chapter arising from translation of it, corrections and edits for house style, removal of problematic material and other reasonable edits.
\n\n3. CORRESPONDING AUTHOR'S DUTIES
\n\n3.1 When distributing or re-publishing the Chapter, the Corresponding Author agrees to credit the Book in which the Chapter has been published as the source of first publication, as well as IntechOpen. The Corresponding Author warrants that each Co-Author will also credit the Book in which the Chapter has been published as the source of first publication, as well as IntechOpen, when they are distributing or re-publishing the Chapter.
\n\n3.2 When submitting the Chapter, the Corresponding Author agrees to:
\n\nThe Corresponding Author will be held responsible for the payment of the Open Access Publishing Fees.
\n\nAll payments shall be due 30 days from the date of the issued invoice. The Corresponding Author or the payer on the Corresponding Author's and Co-Authors' behalf will bear all banking and similar charges incurred.
\n\n3.3 The Corresponding Author shall obtain in writing all consents necessary for the reproduction of any material in which a third-party right exists, including quotations, photographs and illustrations, in all editions of the Chapter worldwide for the full term of the above licenses, and shall provide to IntechOpen upon request the original copies of such consents for inspection (at IntechOpen's option) or photocopies of such consents.
\n\nThe Corresponding Author shall obtain written informed consent for publication from people who might recognize themselves or be identified by others (e.g. from case reports or photographs).
\n\n3.4 The Corresponding Author and any Co-Author shall respect confidentiality rights during and after the termination of this Agreement. The information contained in all correspondence and documents as part of the publishing activity between IntechOpen and the Corresponding Author and any Co-Author are confidential and are intended only for the recipient. The contents may not be disclosed publicly and are not intended for unauthorized use or distribution. Any use, disclosure, copying, or distribution is prohibited and may be unlawful.
\n\n4. CORRESPONDING AUTHOR'S WARRANTY
\n\n4.1 The Corresponding Author represents and warrants that the Chapter does not and will not breach any applicable law or the rights of any third party and, specifically, that the Chapter contains no matter that is defamatory or that infringes any literary or proprietary rights, intellectual property rights, or any rights of privacy. The Corresponding Author warrants and represents that: (i) the Chapter is the original work of themselves and any Co-Author and is not copied wholly or substantially from any other work or material or any other source; (ii) the Chapter has not been formally published in any other peer-reviewed journal or in a book or edited collection, and is not under consideration for any such publication; (iii) they themselves and any Co-Author are qualifying persons under section 154 of the Copyright, Designs and Patents Act 1988; (iv) they themselves and any Co-Author have not assigned and will not during the term of this Publication Agreement purport to assign any of the rights granted to IntechOpen under this Publication Agreement; and (v) the rights granted by this Publication Agreement are free from any security interest, option, mortgage, charge or lien.
\n\nThe Corresponding Author also warrants and represents that: (i) they have the full power to enter into this Publication Agreement on their own behalf and on behalf of each Co-Author; and (ii) they have the necessary rights and/or title in and to the Chapter to grant IntechOpen, on behalf of themselves and any Co-Author, the rights and licenses expressed to be granted in this Publication Agreement. If the Chapter was prepared jointly by the Corresponding Author and any Co-Author, the Corresponding Author warrants and represents that: (i) each Co-Author agrees to the submission, license and publication of the Chapter on the terms of this Publication Agreement; and (ii) they have the authority to enter into this Publication Agreement on behalf of and bind each Co-Author. The Corresponding Author shall: (i) ensure each Co-Author complies with all relevant provisions of this Publication Agreement, including those relating to confidentiality, performance and standards, as if a party to this Publication Agreement; and (ii) remain primarily liable for all acts and/or omissions of each such Co-Author.
\n\nThe Corresponding Author agrees to indemnify and hold IntechOpen harmless against all liabilities, costs, expenses, damages and losses and all reasonable legal costs and expenses suffered or incurred by IntechOpen arising out of or in connection with any breach of the aforementioned representations and warranties. This indemnity shall not cover IntechOpen to the extent that a claim under it results from IntechOpen's negligence or willful misconduct.
\n\n4.2 Nothing in this Publication Agreement shall have the effect of excluding or limiting any liability for death or personal injury caused by negligence or any other liability that cannot be excluded or limited by applicable law.
\n\n5. TERMINATION
\n\n5.1 IntechOpen has a right to terminate this Publication Agreement for quality, program, technical or other reasons with immediate effect, including without limitation (i) if the Corresponding Author or any Co-Author commits a material breach of this Publication Agreement; (ii) if the Corresponding Author or any Co-Author (being an individual) is the subject of a bankruptcy petition, application or order; or (iii) if the Corresponding Author or any Co-Author (being a company) commences negotiations with all or any class of its creditors with a view to rescheduling any of its debts, or makes a proposal for or enters into any compromise or arrangement with any of its creditors.
\n\nIn case of termination, IntechOpen will notify the Corresponding Author, in writing, of the decision.
\n\n6. INTECHOPEN’S DUTIES AND RIGHTS
\n\n6.1 Unless prevented from doing so by events outside its reasonable control, IntechOpen, in its discretion, agrees to publish the Chapter attributing it to the Corresponding Author and any Co-Author.
\n\n6.2 IntechOpen has the right to use the Corresponding Author’s and any Co-Author’s names and likeness in connection with scientific dissemination, retrieval, archiving, web hosting and promotion and marketing of the Chapter and has the right to contact the Corresponding Author and any Co-Author until the Chapter is publicly available on any platform owned and/or operated by IntechOpen.
\n\n6.3 IntechOpen is granted the authority to enforce the rights from this Publication Agreement, on behalf of the Corresponding Author and any Co-Author, against third parties (for example in cases of plagiarism or copyright infringements). In respect of any such infringement or suspected infringement of the copyright in the Chapter, IntechOpen shall have absolute discretion in addressing any such infringement which is likely to affect IntechOpen's rights under this Publication Agreement, including issuing and conducting proceedings against the suspected infringer.
\n\n7. MISCELLANEOUS
\n\n7.1 Further Assurance: The Corresponding Author shall and will ensure that any relevant third party (including any Co-Author) shall, execute and deliver whatever further documents or deeds and perform such acts as IntechOpen reasonably requires from time to time for the purpose of giving IntechOpen the full benefit of the provisions of this Publication Agreement.
\n\n7.2 Third Party Rights: A person who is not a party to this Publication Agreement may not enforce any of its provisions under the Contracts (Rights of Third Parties) Act 1999.
\n\n7.3 Entire Agreement: This Publication Agreement constitutes the entire agreement between the parties in relation to its subject matter. It replaces and extinguishes all prior agreements, draft agreements, arrangements, collateral warranties, collateral contracts, statements, assurances, representations and undertakings of any nature made by or on behalf of the parties, whether oral or written, in relation to that subject matter. Each party acknowledges that in entering into this Publication Agreement it has not relied upon any oral or written statements, collateral or other warranties, assurances, representations or undertakings which were made by or on behalf of the other party in relation to the subject matter of this Publication Agreement at any time before its signature (together "Pre-Contractual Statements"), other than those which are set out in this Publication Agreement. Each party hereby waives all rights and remedies which might otherwise be available to it in relation to such Pre-Contractual Statements. Nothing in this clause shall exclude or restrict the liability of either party arising out of its pre-contract fraudulent misrepresentation or fraudulent concealment.
\n\n7.4 Waiver: No failure or delay by a party to exercise any right or remedy provided under this Publication Agreement or by law shall constitute a waiver of that or any other right or remedy, nor shall it preclude or restrict the further exercise of that or any other right or remedy. No single or partial exercise of such right or remedy shall preclude or restrict the further exercise of that or any other right or remedy.
\n\n7.5 Variation: No variation of this Publication Agreement shall be effective unless it is in writing and signed by the parties (or their duly authorized representatives).
\n\n7.6 Severance: If any provision or part-provision of this Publication Agreement is or becomes invalid, illegal or unenforceable, it shall be deemed modified to the minimum extent necessary to make it valid, legal and enforceable. If such modification is not possible, the relevant provision or part-provision shall be deemed deleted.
\n\nAny modification to or deletion of a provision or part-provision under this clause shall not affect the validity and enforceability of the rest of this Publication Agreement.
\n\n7.7 No partnership: Nothing in this Publication Agreement is intended to, or shall be deemed to, establish or create any partnership or joint venture or the relationship of principal and agent or employer and employee between IntechOpen and the Corresponding Author or any Co-Author, nor authorize any party to make or enter into any commitments for or on behalf of any other party.
\n\n7.8 Governing law: This Publication Agreement and any dispute or claim (including non-contractual disputes or claims) arising out of or in connection with it or its subject matter or formation shall be governed by and construed in accordance with the law of England and Wales. The parties submit to the exclusive jurisdiction of the English courts to settle any dispute or claim arising out of or in connection with this Publication Agreement (including any non-contractual disputes or claims).
\n\nLast updated: 2020-11-27
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