Hypersonic Flow over Closed and Open Nose Missile Bodies: Raw and SVD-Enhanced Schlieren Imaging, Numerical Modeling, and Physical Analysis

Abstract

Understanding and mitigating against high heat loads at leading and blunt aerodynamic surfaces during hypersonic flight represents an ongoing technological challenge. Recent work has shown that the commercial software package, STAR CCM+, can provide reliable predictions of hypersonic aerothermodynamic flow and heating, under a wide range of complex, but common conditions. This chapter presents a preliminary experimental and numerical investigation of hypersonic flow over closed- and open-nose missile bodies, where the latter have been proposed as a means of reducing leading edge heat transfer. Four contributions are presented. First, a novel singular value decomposition (SVD)-based image processing technique is introduced, which significantly enhances the quality of raw schlieren images obtained in high-speed compressible flows. Second, numerically predicted hypersonic flow about a scale-model missile body, obtained using STAR-CCM+, is validated against experimental schlieren image data, an empirical correlation connecting bow shock stand-off distance and shock density ratio, and estimated drag forces. Third, scaling and physical arguments are presented as a means of choosing appropriate gas equations of state and for interpreting results of numerical simulations and experiments. Last, numerical experiments show that the forward facing cavity used in our wind tunnel experiments functions as a heat sink, reducing heat fluxes on the missile body downstream of the cavity.

1. Introduction

Hypersonic flight is often described as flight at free stream Mach numbers exceeding five. However, from a physical standpoint, hypersonic flight refers to conditions in near-body boundary layers where a variety of complex, Mach-number dependent processes can emerge, including non-equilibrium flow of gas constituents, ionization and recombination of gas species, emission and absorption of radiation both within the gas and at the aerodynamic surface, difficult-to-predict laminar-to-turbulent boundary layer flow transition, conversion of high gas enthalpy and kinetic energy into high intensity surface heating, thermal ablation of aerodynamic surfaces, generation of high frequency screech modes, excitation of high frequency in-solid flutter, and various shock-boundary layer interactions.

Predicting and mitigating against intense surface heating and surface stress, particularly at the leading edges of aerodynamic bodies, remains a central challenge in hypersonic aerodynamics. High heat loads and stresses can ablate the leading edge, destabilizing the high speed flow, threatening the integrity of the aerodynamic body, and placing the body well outside its design envelope.

These phenomena are visually represented in Figure 1. As shown, large heat loads exist along all leading edges, being highest at the nose and decaying, due to boundary layer thickening and reduction of viscous heating, with distance from the nose.

Figure 1.

Hypersonic glide vehicle, leading edges experiencing high friction and heat loading [1].

The aerothermodynamic phenomena that arise during hypersonic flight make for very difficult mechanical design. This study focuses on the problem of leading edge heating. Here, a potential approach for mitigating against leading edge heating, introduction of a forward facing cavity, placed at the nose of a missile-shaped test body, is investigated in preliminary fashion.

1.1 Chapter overview and main results

Due to the complexity of hypersonic aerodynamic flows, numerical modeling and experimental diagnostics remain active areas of research. This Chapter presents a preliminary experimental and numerical investigation of heat transfer and flow structure produced by hypersonic flow over a missile shaped body, having both a solid nose and an on-nose forward-facing cavity.

An overview of the work performed and the results obtained is as follows:

1. Hypersonic flow experiments are carried out in the Hypersonic Wind Tunnel facility recently designed and built at UNC Charlotte [2, 3, 4]. Time-dependent schlieren images are obtained for Mach 4.5 flow about a small, solid nosed, missile-shaped body, as well as for Mach 3.5 flow over the same body, modified with an on-nose forward facing cavity.

2. A numerical model simulating these experimental flows is developed using STAR CCM+, a commercial software package that has recently been validated [5] against experimental heat transfer measurements in a range of hypersonic flows, including laminar and turbulent boundary layer flows [5], flows dominated by large separation bubbles [5], and flows featuring both boundary layer-shock interactions [5] and shock-shock interactions [5]. The present study appears to be one of the first to implement the modeling best practices documented in Cross and West’s wide ranging investigation [5]. Following [5], we highlight procedures for setting up high fidelity adaptive grids, setting up grids for resolving turbulent boundary layers, and for diagnosing solution convergence.

3. The Chapter introduces a new technique for significantly enhancing Schlieren images obtained in high speed flows. The technique replaces an array of raw, time-dependent, pixelated Schlieren images with a k-rank singular value image decomposition (SVD) [6], which suppresses static background and enhances time-varying flow features, such as oscillating bow shocks and body-generated Mach waves.

4. The numerical model is validated, in preliminary fashion, via three tests: (i) the computed drag force on the solid nose test article is compared against the drag estimated via Newton’s drag law, (ii) computed schlieren images of the near-body density gradient field are compared against experimental schlieren images, and (iii) computed bow shock standoff distances are compared against a correlation that connects the standoff distance to the cross-shock density ratio.

5. The complexity of hypersonic aerodynamic flows suggests that order of magnitude analyses can play an important role in interpreting experimental results, and in testing the physical consistency of numerical solutions. Here, we use scaling arguments to: (i) explain Newton’s drag law for hypersonic flow [7], and (ii) show that time scales for relaxation of excited vibration modes in shock layer molecular ${\mathrm{N}}_2$ and ${\mathrm{O}}_2$ exceed $\left({\mathrm{N}}_2\right),$ and are of the same order of magnitude $\left({\mathrm{O}}_2\right)$ as the shock layer flow time scale.

2. Experiment: hypersonic flow over solid nose and open nose missile shapes

Between 2019 and 2021, a blow down hypersonic wind tunnel was designed, built and tested at UNC Charlotte [2, 3, 4]; see Figure 2. The wind tunnel facility, pictured in Figure 2, consists of a modular/interchangeable bench top de Laval nozzle, a rectangular $25{\text{in}}^2$ cross-section test section, a diffuser, and near-exit silencer/noise baffle. The test section, 49.5 cm in length, 27.3 cm wide and 25.4 cm in height, has a removable top plate, allowing convenient placement of aerodynamic models within the test section. The test chamber is fitted with a pressure sensor, two thermocouples, and a reinforced static pressure Pitot tube which can be swept across the test chamber inlet. To allow visual access, the lateral sides of the test chamber are fitted with square (15 cm × 15 cm) high strength glass windows.

Figure 2.

UNC Charlotte hypersonic wind tunnel [2, 3, 8].

Compressed room air is stored in twelve high pressure cylinders, each connected to a single high pressure manifold, located immediately upstream of the nozzle plenum. The cylinders, rated for pressures up to 10,000 psi, are charged before each run using a commercial compressor. Prior to any given experimental run, a pressure-actuated, fast response ball valve, separating the manifold and plenum, is closed and manifold pressure is set at a magnitude that produces a pre-specified run time plenum pressure. Plenum/stagnation pressure and temperature are monitored using an in-plenum pressure sensor and thermocouple.

A schlieren system is used to image density gradient fields within the test section; experimental details are available in [3, 4, 8]. For this study, density gradient fields were imaged about a missile body having a solid nose, shown in Figure 3, and an open nose missile body, shown in Figure 4.

Figure 3.

Solid nose model geometry used in wind tunnel experiments and numerical models.

Figure 4.

Open nose/forward facing cavity geometry used in experiments and numerical models.

2.1 Test article geometry

As depicted in Figure 3, the solid nose test piece investigated in this study is a pseudo-blunt body that mimics a missile-shaped body. The total length is 1.35 inches (3.43 cm), the forward diameter is 0.5 inches (1.26 cm), and the aft diameter is 0.6 inches (1.52 cm). The same body dimensions and shape are used in construction of all numerical models. Geometric specifications for the open-nose/forward-facing cavity body are shown in Figure 4. A simple cavity shape was chosen in which the depth and diameter of the circular cavity are the same, 0.2 inches (0.508 cm). Again, these dimensions are used in all numerical models of flow over the open-nose body.

3. Experimental results

Density gradient fields and test chamber pressure histories captured during Mach 4.5 flow over the solid nose missile body are shown in Figure 5. The rapid rise and fall in chamber Mach number is produced by opening and closure of the fast response (0.25 s response time) ball valve separating the high pressure manifold and nozzle plenum. The noisy variations in pre- and post-run Mach numbers reflect leakage of high pressure air from the pressurized manifold through the ball valve into the nozzle plenum.

Figure 5.

Schlieren imaging and Mach number measurements in Mach 4.5 flow over solid nose missile body.

The raw schlieren image clearly indicates the presence of a bow shock, as well as Mach waves emanating from the quasi-cylindrical portion of the missile body. The latter are induced by small grooves in the body surface, produced during machining of the shape. In addition, a (conical) expansion fan is indicated at the trailing (circular) base of the missile body. The somewhat coarse nature of the images, here and below, reflects the nature of these early experiments, designed to shake down and optimize the wind tunnel. Note, that the solid body on the aft side of the test body is a stinger to which the body is attached.

Similar results, shown in Figure 6, are observed for Mach 3.5 flow past the open-nose missile body. Here, the schlieren image is darker and more monochrome than in Figure 5, reflecting slight misalignment of the schlieren system. Due to failure of an early Pitot tube at Mach 4.5, these tests were run at Mach 3.5.

Figure 6.

Schlieren imaging and Mach number measurements in Mach 3.5 flow over open nose missile body.

3.1 Singular value decomposition (SVD) compression for improved schlieren imaging

Singular value decomposition (SVD) has found wide application in image processing [9], where the common theme centers on obtaining a sparse representation, via SV decomposition, of a raw image or a time sequence of raw images, i.e., a video. This study introduces what we believe to be the first application of SVD to the enhancement of raw schlieren images, here obtained in our hypersonic wind tunnel experiments.

The technique, described in [6], proceeds as follows:

1. Raw, digitized schlieren video images, taken during any given experimental run, are sequentially arranged into a time series matrix, $\stackrel{\sim }{A},$ of dimension $M^{\ast }N\times T,$ where each (nominally instantaneous) vectorized digital video frame contains $M\times N$ pixels, and where $T$ is the total number of frames obtained during the experiment.

2. Perform SV decomposition on matrix $\stackrel{\sim }{A}$.

3. Remove static (time-invariant) features from $\stackrel{\sim }{A}$ by performing a rank-k background subtraction,

$$\widehat{A}=\stackrel{\sim }{A}-{\stackrel{\sim }{A}}_k$$E1

where ${\stackrel{\sim }{A}}_k$ is the rank-k approximation of $\stackrel{\sim }{A},$ and where $k\le T\le M^{\ast }N.$ For imaged processes in which a ‘visually dominant’ static background obscures time-varying events, this step suppresses the background, effectively enhancing obscured, time-varying features [6].

Applying this image enhancement technique to our raw schlieren video images, single frames of which are shown in Figures 5 and 6, we obtain the images in Figures 7 and 8. Roughly speaking, in the present application, choosing a larger rank $k$ effectively removes more static background. Comparing Figure 5 with Figures 7 and 8, we observe that an appropriate selection of rank k significantly improves resolution of the density gradient fields extant in our experiments. This feature is particularly apparent in Figure 8, where large scale unsteadiness and turbulence—apparently reflecting high-frequency fluid oscillations in and near the on-nose cavity (see below)—appears to produce an oscillating bow shock, as well as turbulent flow downstream of the shock.

Figure 7.

SVD-enhanced imaging of Mach 4.5 flow over closed nose missile body; left image is a rank 1 representation of a raw schlieren image and the right is a rank 20 representation.

Figure 8.

SVD-enhanced imaging of Mach 3.75 flow over open nose missile body; left image is a rank 1 representation of a raw schlieren image and the right is a rank 20 representation.

Further details and results obtained by this method will be reported in a separate publication.

4. Computational model

Computational models of unsteady, three-dimensional hypersonic flow over the closed and open nose missile bodies used in our experiments were developed. As noted, and based on the recent validation studies reported by Cross and West [5], the commercial CCM+ package was employed. The objectives of this effort were two-fold:

1. Determine appropriate turbulence models and adaptive meshing strategies that provide consistent predictions of experimentally observed density gradient fields, as revealed by schlieren imaging.

2. Use the experimentally validated computational model to investigate, in preliminary fashion, surface heat transfer to closed and open nose missile bodies.

A secondary objective centered on investigating the effects of three gas models – ideal gas, equilibrium real gas, and non-equilibrium, two-specie (nitrogen and oxygen) models – on computed flow fields and surface heat transfer.

In this section, we briefly describe the general flow model, then highlight three modeling features that are crucial to obtaining physically reasonable computed results, and finally discuss a limitation associated with both the present model and the model developed by Cross and West [5].

4.1 Model description

The model solution domain is depicted in Figure 9. The missile body is placed forward of center in a spherical domain. As is often the case in aerodynamic flow models, the size of the solution domain is somewhat arbitrarily chosen; the goal in this study is to make the domain large enough that free stream boundary conditions can be reasonably imposed on the far field boundary. [Model validation against experimental data, here, experimental schlieren images, implies that the chosen domain size is appropriate.]

Figure 9.

Solution domain.

No slip and no penetration conditions are imposed on the missile body surface. Mirror flow (and thermal transport) symmetry is assumed about any plane passing through the center of the sphere. Thus, on the (deep blue) circular symmetry boundary, derivatives of all field variables with respect to the azimuthal angle, $\varphi ,$ are zero. Based on the experimentally validated STAR-CCM+ hypersonic flow simulations in a study conducted by Cross and West [5], the full, variable property Navier-Stokes equations, including the continuity and energy equations, are solved. Technical details concerning model set-up, which followed the best practices outlined in [5], can be found in [8].

4.2 Adaptive meshing and resolution of shocks and turbulent boundary layers

STAR-CCM+ provides a number of utilities that allow high fidelity modeling of stationary hypersonic aerodynamic flows. The most important of these, adaptive meshing, iteratively refines meshes in regions where pressure gradients are high, e.g., in and near shocks. Similarly, in high velocity gradient regions, e.g., turbulent boundary layer viscous sublayers, buffer layers, and logarithmic regions, STAR CCM+ continuously monitors and alters mesh thicknesses.

STAR-CCM+ constructs stationary turbulent solutions of the Navier-Stokes equations in two steps. First, beginning from a specified initial condition and a user-specified initial coarse mesh, STAR CCM+ incrementally increases the free stream Mach number, solves the inviscid Navier-Stokes equations, and, using computed pressure gradient and velocity fields, refines the mesh in high pressure and velocity gradient regions. The initial condition and mesh can be somewhat arbitrarily specified: a global zero or low speed velocity can be imposed, for example, while the initial mesh is constructed via a few user-specified mesh parameters [5]. Examples of inviscid construction of the initial mesh are shown in Figure 10.

Figure 10.

Automated mesh refinement based on inviscid flow solution, obtained at 1000, 3000 and 6000 iterations, respectively; the final inviscid mesh provides the initial mesh for the complete viscous solution.

Once an inviscid solution is obtained at the desired free stream Mach number, STAR CCM+ then iteratively solves the full Navier-Stokes equations, continuing to adapt the mesh as a converged stationary solution is approached. Regarding convergence, in all numerical experiments, we monitor both computed drag force and mesh cell count, stopping a simulation when these have reached nominally steady magnitudes. Figure 11 shows an example.

Figure 11.

Solution convergence is determined by monitoring mesh cell count and net computed drag during iterative solution of the full viscous flow problem.

Cross and West [5] provide detailed guidance on choosing software settings designed to ensure third order solution accuracy (except within shocks, where second order accuracy is achieved), solution stability, and proper resolution of boundary layers.

4.3 Qualitative validation of computational model

Comparing model predictions against experimental data represents the gold standard in code validation. In this study, we are limited to presenting three semi-quantitative checks and one consistency check on our computational model.

First, based on the experimental stagnation temperature and pressures, $T_{01}$ and $P_{01},$ and the missile body geometry used in the model, the predicted drag force on the body, shown in Figure 11, is $D_{\mathit{\text{model}}}\approx 30\mathrm{N}.$ As a rough check on this result, we can estimate the actual drag, $D_{\mathit{est}},$ as follows. First, we note that the pressure at the base of the body is approximately equal to the free stream/test section pressure, $P_{\mathit{\text{base}}}\sim P_1.$ This is shown by recognizing that the flow over the trailing edge of the quasi-cylindrical missile produces a thin shear layer immediately downstream of the body. A (conical) expansion fan emanating from the trailing edge bends the shear layer inward toward the body’s axial centerline. Thus, define a local coordinate system within the shear layer, labeling the cross-layer coordinate as $n,$ and layer-parallel coordinate as $t,$ where $n$ and $t$ are orthogonal. Consider the time-average n-component of the momentum equation. Since the average velocity in the n-direction is negligible, and the turbulent flow is stationary, time-average n-direction inertia and time-average viscous forces are likewise negligible. Thus, the time-average pressure gradient in the n-direction—across the shear layer—is small, on the order of the n-direction gradient in (turbulent) Reynolds stresses: $P_{\mathit{\text{base}}}\sim P_1$.

Next, estimate the drag force on the body as

$$D_{\mathit{exp}}\sim \left(P_1+{\rho }_1{u}_1^2\right)\cdot A-P_{\mathit{\text{base}}}\cdot A$$E2

or

$$D_{\mathit{exp}}\sim P_{01}\cdot A-P_1\cdot A$$E3

where $P_1,$ ${\rho }_1,$ and $u_1$ are the free stream/test section pressure, density, and velocity upstream of the bow shock, and $A$ is the projected area of the missile body (in the direction of the free stream flow). For isentropic flow between the nozzle plenum and upstream face of the bow shock, and for the test section Mach number of 4.5, $P_{01}/P_1\approx 289.$ Thus, $D_{\mathit{est}}\sim P_{01}\cdot A\approx 27\mathrm{N},$ or,

$$D_{\mathit{\text{model}}}\sim D_{\mathit{est}}$$E4

As a second check, and as shown in Figure 12, we compare a computed density gradient field, as indicated by the numerical schlieren image shown, against a corresponding (raw) experimental schlieren image, both obtained for Mach 4.5 flow over the solid nose missile body. Comparing the distances from the bow shock nose to the locations on the upper and lower image boundaries where the bow shock meets the boundary, we find that

Figure 12.

Numerical Schlieren image versus experimentally observed Schlieren for Mach 4.5 flow over solid nose missile body.

$$\frac{\mid L_{\mathit{\text{upper}},\mathit{exp}}-L_{\mathit{\text{upper}},\mathit{\text{model}}}\mid }{L_{\mathit{\text{upper}},\mathit{exp}}}\approx 0.02$$E5

and

$$\frac{\mid L_{\mathit{\text{lower}},\mathit{exp}}-L_{\mathit{\text{lower}},\mathit{\text{model}}}\mid }{L_{\mathit{\text{lower}},\mathit{exp}}}\approx 0.02$$E6

Since computed density gradient fields require accurate solutions for velocity, pressure, temperature, and density fields, this rough comparison indicates the physical fidelity of computed results. Note that the validation case implemented the ideal gas equation of state. As discussed below, improved results would likely be observed using STAR-CCM+’s two-specie nonequilibrium gas model.

As a third check on our model, we compare computed shock displacement distances, ${\mathrm{\Delta }}_{\mathit{bow},\mathit{\text{model}}},$ against an order of magnitude estimate, ${\mathrm{\Delta }}_{\mathit{est}}.$ As shown in Table 1, all three gas models, for both the closed nose and open nose missile bodies, predict ${\mathrm{\Delta }}_{\mathit{\text{model}}}\approx 1\mathrm{mm}.$ A simple order of magnitude estimate for $\mathrm{\Delta }$ follows from a theoretical expression for shock standoff distance adjacent blunt bodies [10, 11, 12]

Geometry	Equation of state	Shock detachment distance (mm)
No cavity	Ideal gas	0.9
	Real gas	1.0
	Thermal non-equilibrium	1.1
Cavity	Ideal gas	1.1
	Real gas	1.2
	Thermal non-equilibrium	1.5

Table 1.

Shock standoff distance as predicted by each equation of state.

$$\frac{\mathrm{\Delta }}{R_{\mathit{\text{nose}}}}=0.82\frac{{\rho }_1}{{\rho }_2}$$E7

where ${\rho }_1$ and ${\rho }_2$ are densities immediately upstream and downstream of the (locally normal) shock, and $R_{\mathit{\text{nose}}}$ is the radius of the missile body nose. Assuming isentropic flow between the nozzle plenum and test section,

$${\rho }_1=\frac{P_{01}}{{\mathit{RT}}_{01}}{\left[1+\frac{k-1}{2}{M}_1^2\right]}^{\frac{-1}{k-1}}$$E8

while ${\rho }_2$ is obtained using the normal shock relations, given ${\rho }_1$ and $M_1$.

Using experimental/model parameters: $P_{01}=1.63\mathrm{Pa},$ $T_{01}=291\mathrm{K},$ and $M_1=4.5$ (for the closed nose body), we obtain

$${\mathrm{\Delta }}_{\mathit{est}}\sim 1\mathrm{mm}$$E9

which, as shown in Table 1, is comparable to $\mathrm{\Delta }$ magnitudes predicted by all three gas models.

Finally, a consistency check on the model’s resolution of the body-adjacent turbulent boundary layer is obtained via plots of the dimensionless thickness, $y_o^{+},$ of the first mesh layer adjacent the body. The parameter $y_o^{+}=y_o/{\delta }_{\mathit{\text{sublayer}}},$ represents the Reynolds number associated with the first mesh layer (located well within the viscous sublayer), where $y_o$ is the dimensional thickness of the first cell, ${\delta }_{\mathit{\text{sublayer}}}=\nu /u^{\ast },$ is the characteristic sublayer thickness, and $u^{\ast }=\sqrt{{\tau }_{\mathit{\text{wall}}}/\rho }$ is the friction velocity. Proper resolution of the viscous sublayer, and by implication, the entire turbulent boundary layer, is indicated by $y_o^{+}$ magnitudes smaller than 1. A typical plot of surface $y_o^{+}$ magnitudes, showing $y_o^{+}<\sim 0.1$ at all locations, is shown in Figure 13.

Figure 13.

Distribution of at-surface viscous sublayer Reynolds numbers, as given by $y_o^{+}$—solid nose missile body.

5. Computational experiments

Experimental test section Mach numbers range from 3.5 to 4.5. Thus, according to [5, 13], molecular vibration modes are collisionally excited in $N_2$ and $O_2$ molecules, both within the test section and downstream of the bow shock. In this section, we first present scaling arguments to: (a) explain the Mach number-vibration nonequilibrium criterion in [5], (b) estimate the time scale, ${\tau }_{\mathit{\text{rlxn}}},$ for relaxation of excited vibration modes, and (c) show that ${\tau }_{\mathit{\text{rlxn}}},$ for $N_2$ is orders of magnitude longer than the (bulk) flow time scale, ${\tau }_{\mathit{\text{flow}}},$ downstream of the bow shock.

Importantly, the scaling arguments indicate that the two-specie $\left({\mathrm{N}}_2\text{and}{\mathrm{O}}_2\right)$ nonequilibrium gas model in STAR CCM+ is the appropriate equation of state—as opposed to available ideal gas and real gas models—for investigating flow and heat transfer about our closed and open nose missile bodies.

5.1 Scaling arguments: physical origin of near-nose high temperature gas region, vibrational excitation of molecular $N_2$ and $O_2,$ and estimated (long) relaxation times

5.1.1 Physical origin of near-nose high temperature gas region

Anderson [13] provides a useful graph indicating temperature ranges over which vibrational excitation, dissociation $\left({\mathrm{N}}_2\to 2\mathrm{N}\text{and}{\mathrm{O}}_2\to 2\mathrm{O}\right),$ and ionization.

$\left(\mathrm{N}\to {\mathrm{N}}^{+}+{\mathrm{e}}^{-}\text{and}\mathrm{O}\to {\mathrm{O}}^{+}+{\mathrm{e}}^{-}\right)$ (in air at 1 atm) take place. Theoretically, minimum, pressure-dependent temperatures marking initiation of these processes can be worked out using molecular collision theory; see, e.g. [14].

As shown in Figure 14, for Mach 4.5 flow over the solid missile body, all three gas models predict similar temperature distributions, with most of the thin gas layer between the shock and body exhibiting temperatures well in excess of 800 K. Physically, there are three possible sources generating this high temperature gas layer:

1. viscous heating within the bow shock, produced by a large reduction in gas velocity—of order ${10}^2\mathrm{m}/\mathrm{s},$ as estimated via simple normal shock theory—over a shock thickness of order of $0.1\mu \mathrm{m}$ [15];

2. viscous heating within the gas layer; and

3. pressure heating, taking place between the upstream face of the bow shock, where $P_1\approx 0.58\mathrm{atm},$ and the missile body nose, where $P=P_{02}\approx 15\mathrm{atm}.$

Figure 14.

Computed temperature distributions for Mach 4.5 flow over solid nose missile body.

Focusing on the energy equation and introducing appropriate length, time, and velocity scales, estimates for temperature increases produced by the first two mechanisms are obtained by balancing the temporal change in fluid enthalpy, against the dominant (streamwise) term in the viscous dissipation function:

$$\rho C_p\frac{\partial T}{\partial t}\sim \mu \frac{{\partial }^{2}u}{\partial x^2}$$E10

Estimating the time scales for mechanisms (a) and (b), respectively, as ${\tau }_{\mathit{\text{shock}}}\sim {\delta }_{\mathit{\text{shock}}}/u_1,$ and ${\tau }_{\mathit{\text{gaslayer}}}\sim \mathrm{\Delta }/u_2,$ we obtain associated estimated temperature increases as follows:

$$\mathrm{\Delta }{T}_{\mathit{\text{viscshock}}}\sim \frac{{\mu }_1}{{\rho }_1{C}_{p1}}\frac{u_1}{{\delta }_{\mathit{\text{shock}}}}\sim 10\mathrm{K}$$E11

$$\mathrm{\Delta }{T}_{\mathit{\text{viscgaslayer}}}\sim \frac{{\mu }_2}{{\rho }_2{C}_{p2}}\frac{u_2}{\mathrm{\Delta }}\sim 0.01\mathrm{K}$$E12

Clearly, viscous dissipation, both within the shock and within the near-body gas layer, plays a minimal role in generating the high temperatures observed between the shock and nose.

By contrast, balancing the temporal enthalpy change (of a fluid particle) against the temporal pressure heating term,

$$\rho C_p\frac{\partial T}{\partial t}\sim \frac{\partial P}{\partial t}$$E13

and again estimating the gas layer (flow) time scale as ${\tau }_{\mathit{gas}\mathit{\text{layer}}}\sim \mathrm{\Delta }/u_2$ (which cancels), we obtain:

$$\mathrm{\Delta }{T}_{\mathit{\text{pressure}}}\sim \frac{P_{02}-P_2}{{\rho }_2{C}_{p2}}\sim {10}^2\mathrm{K}$$E14

where again, from normal shock theory, $P_{02}\approx 15\mathrm{atm}$ and $P_1\approx 0.56\mathrm{atm}.$ Since $\mathrm{\Delta }T=T_{\mathit{\text{surf}}}-T_2,$ where $T_2\approx 300\mathrm{K},$ then the estimated maximum temperature, $T_{\mathit{\text{surf}}},$ observed at the missile body surface, is approximately 400 K. While this is approximately one third to one fourth of the maximum temperature magnitudes observed in our numerical experiments, given the approximate nature of our estimate, we can plausibly argue that pressure heating plays the dominant role in generating near nose, elevated gas temperature and missile surface heat transfer.

A similar analysis helps explain the high temperature gas layer observed in flow over the open nose missile body. See Figure 15.

Figure 15.

Computed temperature distributions for Mach 4.5 flow over open nose missile body.

5.1.2 Slow relaxation of collision-induced molecular vibration modes

Within the near-nose gas layer, all three gas models predict temperatures well in excess of the approximate (1 atm) 800 K threshold [5, 13] where vibration modes in ${\mathrm{N}}_2$ and ${\mathrm{O}}_2$ become significantly excited. Two questions immediately arise:

1. What is the physical origin of the criterion in [5, 13] concerning the temperature-dependent threshold for significant vibrational excitation?

2. Since the mean pressure in the layer is on the order of 10 atm $\left(\sim \left[P_{01}+P_1\right]/2\right),$ how does pressure affect the 800 K excitation threshold?

Since collisions induce vibrational excitation, then considering the equilibrium kinetic energy,

$$e_{\mathit{\text{kinetic}}}=3k_{B}T/2$$E15

and equilibrium vibrational energy [16]

$$e_{\mathit{vib}}=\frac{\mathit{h\nu }}{2}+\frac{\mathit{h\nu }}{exp\left(\mathit{h\nu }/k_{B}T\right)-1}$$E16

of individual ${\mathrm{N}}_2$ and ${\mathrm{O}}_2$ molecules at temperature $T,$ then for $T=800\mathrm{K},$

$$e_{\mathit{\text{kinetic}}}\left(800\mathrm{K}\right)\approx e_{\mathit{vib}}\left(800\mathrm{K}\right)/2$$E17

where $k_B,$ $h,$ and $\nu$ are, respectively, Boltzmann’s constant, Planck’s constant, and the ground state vibration frequency. Thus, in an order of magnitude sense, the 800 K threshold corresponds to the temperature at which kinetic energy is high enough to excite significant vibration.

Considering the second question, since $e_{\mathit{\text{kinetic}}}$ and $e_{\mathit{vib}}$ only depend on $T,$ while the collision frequency is proportional to pressure, ${\nu }_{\mathit{col}}\propto P/\sqrt{T}$ [5], we see that the $800\mathrm{K}$ excitation threshold does not significantly shift in the high temperature/elevated pressure gas layer.

5.1.3 Slow vibrational relaxation of ${\mathrm{N}}_2$ and $O_2$ in the near-nose region

Considering relaxation of vibrationally excited ${\mathrm{N}}_2$ and ${\mathrm{O}}_2$ within and downstream of the high temperature near-nose gas region, the Landau-Teller model of vibrational relaxation [17], provides a relatively straightforward description. In order to estimate the time scale, ${\tau }_{\mathit{\text{rlxn}}},$ on which excited molecules relax to a new, elevated equilibrium temperature, the following empirical relationship can be used [13]:

$${\tau }_{\mathit{\text{rlxn}}}=\frac{C_1}{P}exp{\left(C_2/T\right)}^{1/3}$$E18

where, for ${\mathrm{N}}_2$ and ${\mathrm{O}}_2,$ the empirical constants are given by: $C_{1,N_2}=7.12\times {10}^{-3}\mathrm{atm}-\mu \mathrm{s},$ $C_{1,O_2}=5.42\times {10}^{-5}\mathrm{atm}-\mu \mathrm{s},$ $C_{2,N_2}=1.91\times {10}^6\mathrm{K},$ and $C_{2,O_2}=2.95\times {10}^6\mathrm{K}.$ Using $P\sim P_{02}\approx 15\mathrm{atm},$ $T\sim {10}^3\mathrm{K}$ (based on numerically observed near-nose temperatures), approximate relaxation times are:

$${\tau }_{\mathit{\text{rlxn}},N_2}\sim {10}^2\mu \mathrm{s}{\tau }_{\mathit{\text{rlxn}},O_2}\sim 10\mu \mathrm{s}$$E19

Importantly, in the high temperature, near-nose gas region, the relaxation time for ${\mathrm{N}}_2,$ comprising $78\%$ (mole fraction) of air, is an order of magnitude longer than the bulk flow time scale, ${\tau }_{\mathit{\text{flow}}}={\tau }_{\mathit{gas}\mathit{\text{layer}}}\sim \mathrm{\Delta }/u_2,$ where $u_2\sim u_1{\rho }_1/{\rho }_2\approx 1.2\mathrm{\Delta }/R_{\mathit{\text{nose}}}$ Eq. (7), or

$${\tau }_{\mathit{\text{flow}}}\sim 10\mu \mathrm{s}$$E20

Physically, while ${\mathrm{O}}_2$ relaxes relatively quickly – ${\tau }_{\mathit{\text{rlxn}},O_2}\sim {\tau }_{\mathit{\text{flow}}}$ – ${\mathrm{N}}_2$ relaxes much more slowly, ${\tau }_{\mathit{\text{rlxn}},N_2}\sim 10\times {\tau }_{\mathit{\text{flow}}},$ so that significant vibrational energy, again excited by pressure heating between the bow shock and nose, is transported downstream of the nose, consistent with the predicted gas temperature distributions in Figure 14.

Finally, and importantly, the close qualitative and quantitative agreement observed between predicted temperature distributions in Figures 14 and 15, provides strong evidence that the empirical relations introduced in STAR CCM+’s ideal and real gas models [5] are physically reasonable.

5.2 Boundary layer transition and a near-nose ‘ring of fire’

Our numerical experiments indicate that within approximately $50\mathrm{\mu m},$ of the solid nose, a rapid laminar to turbulent boundary layer transition takes place. Evidence of transition, predicted by all three gas models, is captured in Figures 16 and 17. Focusing on the nonequilibrium model prediction in Figure 16, a rapid, micron-scale drop in near-nose surface heat flux is followed by a slower, millimeter-scale rise. Similar qualitative behavior, although less apparent, is also predicted by the ideal and real gas models. As shown in Figure 17, transition produces a nominally circular, localized region of elevated surface heat fluxes. The nonequilibrium and real gas models predict comparable maximum heat flux magnitudes within this zone, while the maximum flux predicted by the ideal gas model is approximately 70 to 75% of these. Interestingly, and likely reflecting the presence of nonequilibrium ${\mathrm{N}}_2,$ the nonequilibrium gas model exposes a highly localized, on-nose, heat flux approximately 50% larger than off-nose $q_{\mathit{max}}$ magnitudes (predicted by the nonequilibrium and real gas models). At locations downstream of the (hemispherical) nose shoulder, all three models predict similar, relatively low magnitude, nominally fixed heat flux distributions. For reference, near nose predicted surface fluxes are comparable to those measured during atmospheric reentry of lunar mission Apollo capsules during the 1970s [18].

Figure 16.

Boundary heat flux distribution for each gas equation of state—Mach 4.5 flow over solid missile body.

Figure 17.

Boundary heat flux distribution for each gas equation of state—Mach 4.5 flow over solid missile body—three-dimensional view.

Interestingly, experimentally observed heat flux distributions on solid nose projectile bodies in [19]—having similar shapes as those used in the present study—were too (spatially) coarse to resolve both the near-nose turbulent boundary layer transition and the resulting ring of intense heat transfer observed here.

The physical origin of the rapid transition to the near-nose turbulent boundary layer, taking place on a ten micron length scale, remains an open question. Based on observed subsonic conditions within the hemispherical gas layer of radius $\sim R,$ and thickness $\sim \mathrm{\Delta },$ between the bow shock and nose—see Figure 18—we postulate that an acoustic feedback mechanism underlies the fast transition. Specifically, upstream-going acoustic disturbances, generated by the near-nose turbulent boundary layer, generate random, small amplitude oscillations in the bow shock. The latter, in turn, generate down-stream going pressure and density waves that drive the fast transition. We note that unsteady bow shocks in hypersonic flow over axisymmetric cone-tipped cylindrical bodies have recently been observed [20]. Intriguingly, in that study, shock oscillations required an unsteady separation bubble and an unstable shear layer passing over the bubble. In our numerical experiments, the unsteady subsonic hemispherical layer may play an analogous role in driving presumed oscillatory shock dynamics.

Figure 18.

Computed Mach number fields for Mach 4.5 flow over closed and open nose missile shapes.

5.3 Heat sink effect of forward facing cavities

Our numerical experiments reveal that along the cylindrical walls, as well as the circular base of our forward facing cavity, surface heat fluxes exceed the maximum flux observed on the solid nose body; compare Figures 16 and 19. The free stream Mach number in both cases is 4.5; since the ideal gas and real gas models appear to under-predict in-cavity heat transfer—see [8]—the results shown in Figure 19 are obtained using the two-specie non-equilibrium gas model.

Figure 19.

Computed boundary heat flux distribution for Mach 4.5 flow over open nose body; two-species thermal non-equilibrium gas model.

We surmise that concentration of thermal energy within the cavity is produced by relatively low in-cavity velocities—see Figure 20—which extend residence/flow time scales, thus enhancing deposition of decaying vibrational energy. As discussed in [8], and in contrast to the quantitatively consistent flux predictions for flow over the cavity-free body—see Figure 16—STAR CCM+’s ideal and real gas models apparently under-predict this effect when the ratio of flow to (vibration) relaxation time scale becomes too large.

Figure 20.

Computed instantaneous in-cavity and near-nose velocity fields for Mach 4.5 flow over open nose body; two specie, non-equilibrium gas model.

As shown in the numerical experiments in [8], and consistent, for example, with the numerical simulations in [21], introduction of a forward facing cavity reduces heat transfer to the aerodynamic body at all points downstream of the cavity. Physically, this appears to reflect the cavity functioning as a heat sink, transferring gas enthalpy to the walls of the cavity. Clearly, this heat sink effect must be accommodated for in designing open nose hypersonic aerodynamic bodies.

6. Conclusions

1. Schlieren images of Mach 4.5 and 3.5 flow, about closed and open nose missile bodies, respectively, were obtained in the hypersonic wind tunnel at UNC Charlotte.

2. Comparisons between numerically predicted and experimentally observed as well as between numerically calculated and physically estimated drag forces and shock standoff distances suggest that the STAR CCM+ software package provides high quality predictions of hypersonic aerodynamic flows, consistent with the results reported in [5].

3. A new technique, based on static background removal and SVD decomposition of raw digital schlieren images, provides significantly enhanced detail of experimentally imaged high speed flow fields.

4. Numerical experiments reveal that in hypersonic (Mach 4) flow over solid nose bodies, turbulent boundary layer transition takes place on tens of micron length scales. The physical origin of this extremely rapid transition remains an open question. Near-nose transition produces a ring-shaped zone of enhanced heat transfer immediately downstream of the missile nose.

5. Numerical experiments also reveal that forward facing cavities function as heat sinks, absorbing elevated near-nose gas enthalpy and reducing heat transfer to the body downstream of the cavity.