Bifurcation and Instabilities in Fluid Flow

Abstract

In some situations, pipe layout and incoherent behavior of the flow is detected. An instability is generated, and the flow jumps between the values in flow rate or pressure, with no explanation. Usually, the industry solves the problem by creating an exclusion area. It is very important to understand those instabilities, named bifurcation, and why the Navier-Stokes equation has two equiprobable solutions. With this knowledge, the situation can be corrected, and the problem solved. The result is an increase in performance, reliability, etc. and then in the economy of the process. The bistable flow in boiling water reactor (BWR) nuclear power plant is a clear example.

1. Introduction

There are many references in the fluid dynamics literature to typical fluid flow instability problems. Some of the existing examples show the existence of pulsating and oscillating anomalies when they should not be present [1, 2, 3, 4, 5]. These problems led to suspicions of anomalous solutions to the Navier-Stokes equation [5, 6].

The aim of this work is the determination of instabilities in fluid flow in piping systems. These instabilities will be classified as bifurcations since they constitute situations of two valid solutions of the Navier-Stokes equation. For the specific case, the pipe system is a symmetrical system composed of a riser, a manifold, and five symmetrically arranged rising branches.

These real events in fluid flow in pipes, normally produce a reaction in the operators of the systems, since their existence results in problems of production [7], reliability [8] of components that lead to decisions, which redound in the economy of the productive processes.

Thus, we are faced with systems that, from a certain value of one of their parameters, oscillate between two states. For example, from a given flow rate value (position of a control valve) the system oscillates between two pressures, or for a pressure value, the system has two flow rate values.

Normally, hydraulic studies are carried out on the basis of dimensionless numbers, the Reynolds number being the most commonly used. In this study, because the geometry is fixed, the fluid is in isothermal conditions, and the flow rate, flow velocity, or Reynolds number is equivalent. Likewise, since both flow rate and fluid velocity depend only on the position of the control valve, we can also make the equivalence between Re and valve position.

Finally, it must be said that this study is the end of a line of work, starting with the search for a solution to the problem of pulsations in the recirculation system of a boiling water reactor. The initial studies, which are empirical [9, 10, 11], have already determined the solution. Years later, CFD studies corroborated the empirical solutions and provided new ones [12, 13, 14].

Once all this modeling and analysis work has been completed, it is proposed to determine the true root cause, in order to improve the possibility of other solutions or even the explanation of other effects. This is the aim of the present work.

2. What a bifurcation is

Suppose in a dynamic system, a bifurcation occurs when a small smooth change made to the parameter values of the system, causes a sudden qualitative or topological change in its behavior. The name of bifurcation was first introduced by Henry Poincare [15] in 1885, in the first paper in mathematics showing such behavior. Bifurcation occurs in both continuous systems described by Ordinary Derivative Equations (ODE’s), Differential Derivative Equation (DDE’s), and Partial Derivative Equations (PDE’s) and discrete systems described by maps.

The previous paragraph is a qualitative description of what a bifurcation is, and the next will be a more detailed and mathematical description of it.

The explanation or justification of a bifurcation will start with the concept of the asymptotic solution to an evolution problem. An evolution problem has the form shown in Eq. (1).

$$\frac{\mathrm{dU}}{\mathrm{dt}}=\mathrm{F}\left(\mathrm{t},\mathrm{\mu },\mathrm{U}\right)$$E1

where t ≥ 0 is the time and μ is a parameter that lies on the real line, −∞ ≤ μ ≤ ∞. The unknown is U(t) and F(.,.,.) is a given function, i.e.: Partial derivative Equation, ordinary derivative equation. When F() is non-time-dependent, F=F(μ,U), the evolution of U(t) is governed by its initial value U(0) = U₀.

An asymptotic solution is defined as the solution to which U(t) evolves after the transient effects associated with the initial value. Asymptotic has two main types: steady solutions and T-periodic solutions for non-autonomous problems.

Once the problem is enunciated, then the bifurcation definition is possible. Bifurcation solutions are asymptotic solutions that form intersecting branches in suitable space of functions. For example, when U lies in R, the bifurcating steady solution form intersecting branches of the curve F(μ,U) = 0 in the μ,U plane.

So, an asymptotic solution bifurcates from another at μ = μ₀ if there are two distinct asymptotic solutions U⁽¹⁾(μ,t) and U⁽²⁾(μ,t) of the evolution problem continuous in μ and such that U⁽¹⁾(μ₀,t) = U⁽²⁾(μ₀,t).

Note that not all asymptotic solutions arise from bifurcation, there are other solutions such as isolated solutions or disjoint solutions.

From this point, a one-dimensional problem will be the real problem Eq. (1), could be rewritten as:

$$\frac{\mathrm{du}}{\mathrm{dt}}=\mathrm{F}\left(\mathrm{\mu },\mathrm{u}\right)$$E2

where F(.,.) has two continuous derivatives with respect μ and u. Another assumption is that:

$$\mathrm{F}\left(\mathrm{\mu },\mathrm{0}\right)=\mathrm{0}.\forall \mathrm{\mu }\in \mathrm{R}$$E3

The equilibrium solution of Eq. (1) in u = ε satisfied that:

$$\mathrm{F}\left(\mathrm{\mu },\mathrm{\epsilon }\right)=\mathrm{0}$$E4

The study of bifurcation is the study of the equilibrium evolution given by Eq. (4) in the plane (μ,ε).

Theorem:

Let F(μ₀,ε₀) = 0 and let F be continuously differentiable in some open region containing the point (μ₀,ε₀) of the (μ,ε) plane. Then, if F_ε(μ₀,ε₀) ≠ 0 there exists α,β > 0 such that:

1. The Eq. F(μ,ε) = 0 has a unique solution ε = ε(μ) when μ₀-α < μ < μ₀₊α such that ε₀-β < ε < ε₀₊β.

2. The function ε(.) is continuously differentiable when μ₀-α < μ < μ₀₊α

3. ε_μ(μ) = -F_u(μ,ε(μ))/ F_ε(μ,ε(μ))

The solutions of the equilibrium could be the following points:

• Regular point

• Regular turning point

• Singular point

• Double point

• Singular turning (double) point

• Cusp point

• Conjugate point

• High order singular point

In this chapter, the focus will be on the double point. A double point is a point of the curve F(μ,ε) = 0 through which pass two and only two branches of F(μ,ε) = 0 possessing distinct tangents. It is assumed that all second derivatives of F(.,.) do not simultaneously vanish at a double point.

Then, equilibrium curves passing through the singular points satisfy:

$$\mathrm{2}\mathrm{F}\left(\mathrm{\mu },\mathrm{\epsilon }\right)={\mathrm{F}}_{\mathrm{\mu \mu }}\mathrm{\delta }{\mathrm{\mu }}^{\mathrm{2}}+\mathrm{2}{\mathrm{F}}_{\mathrm{\epsilon \mu }}\mathrm{\text{δεδμ}}+{\mathrm{F}}_{\mathrm{\epsilon \epsilon }}\mathrm{\delta }{\mathrm{\epsilon }}^{\mathrm{2}}+\mathrm{o}\left[{\left(\left|\mathrm{\delta \mu }\right|+\left|\mathrm{\delta \epsilon }\right|\right)}^{\mathrm{2}}\right]=\mathrm{0}$$E5

In the limit (μ,ε)⟶ (μ₀,ε₀) for the curves F(.,.) = 0 is reduced to:

$${\mathrm{F}}_{\mathrm{\mu \mu }}\mathrm{d}{\mathrm{\mu }}^{\mathrm{2}}+\mathrm{2}{\mathrm{F}}_{\mathrm{\mu \epsilon }}\mathrm{\text{dεdμ}}+{\mathrm{F}}_{\mathrm{\epsilon \epsilon }}\mathrm{d}{\mathrm{\epsilon }}^{\mathrm{2}}=\mathrm{0}$$E6

For the tangents to the curve. Solving Eq. (6) the Eq. (7) is obtained.

$$\left[\begin{array}{c}{\mathrm{\mu }}_{\mathrm{\epsilon }}^{\mathrm{1}}\left({\mathrm{\epsilon }}_{\mathrm{0}}\right)\\ {\mathrm{\mu }}_{\mathrm{\epsilon }}^{\mathrm{2}}\left({\mathrm{\epsilon }}_{\mathrm{0}}\right)\end{array}\right]=-\frac{{\mathrm{F}}_{\mathrm{\epsilon \mu }}}{{\mathrm{F}}_{\mathrm{\mu \mu }}}\left[\begin{array}{c}\mathrm{1}\\ \mathrm{1}\end{array}\right]+\sqrt[\mathrm{2}]{\frac{\mathrm{D}}{{\mathrm{F}}_{\mathrm{\mu \mu }}^{\mathrm{2}}}}\left[\begin{array}{c}\mathrm{1}\\ -\mathrm{1}\end{array}\right]$$E7

where

$$\mathrm{D}={\mathrm{F}}_{\mathrm{\epsilon \mu }}^{\mathrm{2}}-{\mathrm{F}}_{\mathrm{\mu \mu }}{\mathrm{F}}_{\mathrm{\epsilon \epsilon }}$$E8

Then (μ₀,ε₀) is a double point if D > 0 additionally to have two tangents with slopes different from zero, the condition has to be F_μμ ≠ 0.

3. Bifurcation in fluid flow

The Navier-Stokes´ equation is a PDE, so it is potentially affected by bifurcation as explained in previous paragraphs. Additionally, the Navier-Stokes´ equation is nonlinear so the occurrence of bifurcation is much more probable.

Now the concept of bifurcation, as described before, will be developed mathematically for fluid flow problems. First, some assumptions have been considered:

• Fluid is in an incompressible state.

• Fluid is at constant temperature.

• The part of the system under study has constant section (pipe).

• There’s no fluid sumps or sources.

In this situation, the equation could be written as shown in Eq. (9).

$$\frac{\mathrm{D}\overrightarrow{\mathrm{u}}}{\mathrm{Dt}}=\overrightarrow{\mathrm{F}}-\nabla \mathrm{P}+\frac{\mathrm{\mu }}{\mathrm{\rho }}\left(\frac{1}{3}\nabla \left(\nabla \overrightarrow{\mathrm{u}}\right)+{\nabla }^2\overrightarrow{\mathrm{u}}\right)$$E9

The equation has the following vectors: u (speed), P (pressure), μ (viscosity), and ῤ (fluid density). Since the system has a constant section and there are no fluid sumps or sources, integrated fluid speed is like the scalar flow value. Other supposition is that the system status will depend on a given parameter “A,” for example pump speed, control valve position, Pressure regulation vale position, etc. In this situation, the Navier-Stokes´ equation can be reformulated as:

$$\frac{\mathrm{dq}}{\mathrm{dt}}=\mathrm{F}\left(\mathrm{q},\mathrm{A}\right)$$E10

where q = flow and A = system parameter.

Primary derivatives of Eq. (10) are determined by Eq. (11), with notation and secondary derivatives being simplified in Eq. (12):

$$\frac{\mathrm{\partial }\mathrm{F}\left(\mathrm{q},\mathrm{A}\right)}{\mathrm{\partial }\mathrm{q}}={\mathrm{F}}_{\mathrm{q}}\mathrm{y}\frac{\mathrm{\partial }\mathrm{F}\left(\mathrm{q},\mathrm{A}\right)}{\mathrm{\partial }\mathrm{A}}={\mathrm{F}}_{\mathrm{A}}$$E11

$$\frac{\mathrm{\partial }{\mathrm{F}}^2\left(\mathrm{q},\mathrm{A}\right)}{{\mathrm{\partial }\mathrm{q}}^2}={\mathrm{F}}_{\mathrm{qq}};\frac{\mathrm{\partial }{\mathrm{F}}^2\left(\mathrm{q},\mathrm{A}\right)}{\mathrm{\partial }{\mathrm{A}}^2}={\mathrm{F}}_{\mathrm{AA}}\mathrm{y}\frac{\mathrm{\partial }{\mathrm{F}}^2\left(\mathrm{q},\mathrm{A}\right)}{\mathrm{\partial }\mathrm{qA}}={\mathrm{F}}_{\mathrm{qA}}$$E12

The stability analysis results from establishing a null time derivative. In other words, if the system does not change and A do not vary, flow remains the same. Mathematically, this is seen in Eq. (13).

$$\frac{\mathrm{dq}}{\mathrm{dt}}=0=\mathrm{F}\left(\mathrm{q},\mathrm{A}\right)$$E13

That means analyzing bifurcations in Eq. (11) is the same as analyzing specific F curve points (q, A) on the A parameter-flow plane. Considering these parameters, the points that confirm the Eq. (13) are considered unique and can be classified as follows: regular points, regular inflection points, unique points, double curve point, double inflection points, and peak points.

The double curve point generates two solutions and a curve running through the unique point that has two slopes. Although there are multiple potential shapes, the one selected provides two stable results as it is coherent with bistable conditions. Once shape is selected, it can have three main sub-types, as seen in Figure 1: supercritical, subcritical, and transcritical.

Figure 1.

Sub-types of double point bifurcation (pitchfork type).

In the case of double point bifurcation, there is a balance point with two curves and two different slopes [16]. Curve tangents conform to Eq. (14)

$$\left[\begin{array}{c}\frac{{\mathrm{dq}}^{1)}}{\mathrm{dA}}\\ \frac{{\mathrm{dq}}^{2)}}{\mathrm{dA}}\end{array}\right]=\left[\begin{array}{c}{\mathrm{q}}_{\mathrm{A}}^{1)}\\ {\mathrm{q}}_{\mathrm{A}}^{2)}\end{array}\right]=-\frac{{\mathrm{F}}_{\mathrm{qA}}}{\mathrm{Fqq}}\left[\begin{array}{c}1\\ 1\end{array}\right]-\sqrt[2]{\frac{\mathrm{D}}{{\mathrm{F}}_{\mathrm{qq}}}}·\left[\begin{array}{c}1\\ -1\end{array}\right]$$E14

where “D” is determined by Eq. (15):

$$\mathrm{D}={\mathrm{F}}_{\mathrm{qA}}^2-{\mathrm{F}}_{\mathrm{AA}}·{\mathrm{F}}_{\mathrm{qq}}$$E15

Slope analysis results from analyzing the value of parameter D. If D < 0, there are no real tangential lines on the point, which means the existence of a double point and two slopes can only be justified when D > 0. In this case, if specific concepts are renamed, flow curve slopes in relation to A value are determined by Eq. (16):

$$\frac{\mathrm{dq}}{\mathrm{dA}}\left({\mathrm{A}}_0\right)={\mathrm{q}}_{\mathrm{A}}\left({\mathrm{A}}_0\right)$$E16

For this equation to be true and conformed to, the condition of F_qq not being null must be satisfied, which is exactly what happens as this is a process related to fluid flow. The other condition for conformance to D > 0 is the verification of Eq. (17):

$${\mathrm{F}}_{\mathrm{qA}}>\sqrt[2]{{\mathrm{F}}_{\mathrm{AA}}·{\mathrm{F}}_{\mathrm{qq}}}$$E17

To conclude this theoretical development, the requirements for existence of a double point bifurcation are mathematically conformed because F_qq is not null, and D can under specific conditions be higher than zero. And as the solution existence theorem says, if F(q, A) is continuous, then at least one solution of the Eq. (9) exists.

4. A real case of bifurcation in fluid flow. Bistable flow

In 1985, an electrical power fluctuation was detected at the Leibstadt Nuclear Power Plant during startup and load tests at 100%. Event analyses revealed additional fluctuations in steam flow, thermal power, core flow, and recirculation loop flowrates. Loop “A” fluctuation ranged between 2.5% and 3%, whereas in loop “B” it was between 3% and 3.5%. It was concluded that recirculation flow fluctuations were caused by a bistable flow pattern in the pump discharge header.

In 1986, the first regulatory reference on abnormal performance of recirculation loop flows was written [17]. This document recaps events, like that of Leibstadt, at the USA stations of Pilgrim (1985) and Vermont Yankee (1986), and also determines that fluctuations vary from station to station and even within the loops of one single unit. In accordance with the information available at the time, the NRC establishes that magnitude and duration are not predictable using the analytical methods applied to the piping systems under analysis.

In 1988, General Electric, the responsible (OEM) for the design of the recirculation loop, issued a letter [9] establishing that the problem affects Boiling Water Reactors (BWR) from generation 3 to 6. More specifically, the problem is detected at the collectors of feed pipes supplying jet pumps (Figure 2). The letter was later revised in 2006 to include operational experiences and operating recommendations.

Figure 2.

Recirculation loop schematics in a generation 6 BWR.

Around that time, the same phenomenon occurred in Japan, a country with many boiling water reactors. Since 1986 to 1989, a group of researchers applied various hydraulic models to replicate [10], characterize [11], and propose compensatory measures [18]. Eventually, the feed pipe header of jet pumps was successfully modified.

As this issue was considered non-safety related, bistable flow was rendered acceptable at the plants and unit operation took it into account. At times, operational strategies were modified to minimize this phenomenon, although its analysis did not go any further.

In 2006, a bistable flow phenomenon analysis was presented at Laguna Verde nuclear power plant in México [19].

After 2008, new research trends focused on providing a more detailed bistable flow profile [12], concluded that it can be described as a noise-induced transition mechanism [20]. In this theory, noise is identified as flow turbulence under high Reynolds values.

In 2009, bistable flow analyses continued using mathematical techniques such as wavelets [21] and codes for fluid mechanics. These were used to replicate the results of hydraulic models from the 80s [13].

Ever since they were used in nuclear power plants, bistable flow analyses have identified jet pumps as the most critical hydraulic element. In the year 2011, an analysis was undertaken to determine potential bistable flow impact on jet pumps. This analysis [14] presented the spectral situation of jet pumps, which were influenced by resonance frequencies and induced frequencies resulting from bistable flow. Also, in 2011, a new 3D CFD model confirmed the existence of two states and the non-convergence of the stable condition at specific Reynolds values. It was verified that current lines derived from 3D model results are coherent with hydraulic model data and that geometry changes contribute to eliminating this phenomenon as the stable state is quickly reached and calculation converge is ensured. The fact that for similar environment conditions, there were two solutions for the Navier-Stokes equation suggests the existence of a bifurcation.

Operationally speaking, a few works were undertaken to map out recirculation loop flows in relation to reactor power. The aim was to identify the area most prone to developing bistable flow for several reasons, including flowrate [7].

In line with the above mentioned in this section, the state of the art in bistable flow can be summarized as follows:

• Unpredictable, fluctuating hydraulic phenomenon.

• Hydraulic phenomenon only removable at this point through physical hydraulic circuit modifications.

• Phenomenon characterized by high turbulence which at a certain level leads to a bistable state.

• Phenomenon not impacting plant safety.

The methodology used in this study has two main phases.

Phase 1. Analyze recorded recirculation flow and FCV position data for a full 24-month cycle at an operating station. From this point on, the value of the FCV position will be assimilated to the value of “A” in Eqs. (4) and (7)-(11).

• S1. Determines the curve for recirculation flow (%) vs. FCV position (%).

• S2. Establishes the calculated curve adjustment error based on available data. Analyze error behavior in relation to FCV position (%). The analysis of this error, starting point of the bifurcation study, provides a clear picture of bistable flow.

Phase 2. Determine, find, and characterize the mathematical bifurcation physically supporting bistable flow. Apply theoretical results from the previous section to the time series. Considering the noise nature of the series and the difficulty to determine analytical expressions, an empirical approach is developed. The global interpretation includes the definition of bifurcation maps and their characteristics.

4.1 Analysis (phase 1)

FCV position (%) and flow (%) are interrelated time series [22, 23]. It is also important that time series, especially those for flow, are highly noise signals and will require the use of some mathematical techniques [24].

The analysis of recirculation loop flow signal has several phases aimed at determining flow signal features. Once this information is obtained, the entire system is analyzed, with a special focus on the relationship between recirculation loop flow and FCV opening. The limits of bistable occurrence and ideas suggesting bifurcation existence, mostly as a result of error analyses, will appear at this point.

4.1.1 Opening-flow curve analysis

A Boiling Water Reactor (BWR) 6 reactors have two recirculation loops, each with its own flow control valve (FCV) (Figure 3). Figure 3 shows the point cloud generated from loop flow (%) and y FCV position (%) records during reference plant operation over a 24-month operating cycle.

Figure 3.

Plot of operational points (flow % vs. FCV opening %).

A morphological analysis of the point cloud reveals that after a 60% opening approximately, the cloud begins to widen and after 62% it has a new width which remains constant until the end. This is seen in Figure 4.

Figure 4.

Details of operational points. (FCV opening >55%).

The system designer and technologist determined, both mathematically and theoretically, a polynomial of degree 3 for the flow equation based on FCV opening. This adjustment, with a confidence level of 99.999%, provides the coefficients shown in Table 1.

	${\text{Flow}}\left(\%\right)=a\ast {\left({FC}{V}_{{POS}}\right)}^3+b\ast {\left({FC}{V}_{{POS}}\right)}^2+c\ast \left({FC}{V}_{{POS}}\right)+d$
	a	b	c	d
Design adjustment	4·10^−5	−0.0149	1.9502	21.191
Loop B adjustment	6.5801·10^−5	−0.0188	2.1469	18.5307

Table 1.

Adjustment polynomial coefficients for the theoretical curve adjusted with actual data.

Table 1 reveals the coherence between the theoretical equation and that adjusted with actual data.

4.1.2 Analysis of error adjustment (S2)

Adjustment error (Eq. (18)) is established as the difference between the measured flow value (%) and the adjusted curve value, as seen in Table 2 parameters. Globally speaking, throughout the entire FCV position range there is an error, as seen in Figure 5.

Lambda	Tau (−)	Tau (+)	Reference figure
−32	−0.577	−0.577
0	0	0	Figure 5
1.45	−0.674	0.293
1.76	−0.675	0.294	Figure 6
3.75	−0.728	0.306
5.95	−0.785	0.313
9.55	−0.863	0.256	Figure 7
15.25	−0.955	0.287
19.75	0	1.3	Figure 8

Table 2.

Maximum tau and lambda values.

Figure 5.

Adjustment error.

$$\mathrm{\text{Error}}\left(\mathrm{\text{FCVOpening}}\right)=\mathrm{\text{MeasuredFlow}}\left(\%\right)-\mathrm{\text{CalculatedFlow}}\left(\%\right)$$E18

Figure 5 confirms the idea that there is phenomenon changing flow patterns when FCV opening is 60% or above. This is indicated by the error value, which changes in shape and values. Error varies from about 0% with a + −0.5% band to a + −1.5% band. Even for values exceeding 80%, there is a clearly strong, differentiated bias with regards to flow behavior at this FCV opening value.

4.2 Bifurcation determination (phase 2)

The existence of a bifurcation structure will be demonstrated in this phase through the selection of FCV opening as the bifurcation parameter as it is the only one of a variable nature in the system.

This section focuses on characterizing the pitchfork-type bifurcation for recirculation flow under bistable conditions. For clarity purposes and without losing accuracy, the unique point of double slope will be (0,0) on the plane (FCV opening, flow).

The coordinate parameter, known as tau (τ), will be the error included in Eq. (12), whereas the abscissa, known as lambda (λ), will be the difference between FCV position and the initial bistable point. Thus, considering an opening value of 62.3% as the initial bifurcation point, variable λ will be the result of subtracting from FCV opening (%) the previous value of 62.3%. In other words, initial bifurcation will be λ = 0 and τ = 0.

Instead of interpreting the signal as a whole (FCV opening-flow), recirculation loop flow analysis will be based on various FCV opening levels. The values selected for this task are those in which FCV opening (5) is 29, 62, 62.3, 63, 64, 68, 72, 77, and 82%. These values will be reduced to parameter λ, as see on Table 2. The average value for reduced tau flow was selected when there was only a single peak in the normal distribution or two maximum values in two normal distributions due to bistable flow. These tau values can be seen in Table 2.

The graphical representation of values in Table 2 is seen in Figures 6–9. These values are used to create a graph considered the bifurcation map (Figure 10).

Figure 6.

Recirculation flow evolution with lambda value at 0.00.

Figure 7.

Recirculation flow evolution with lambda at 3.8.

Figure 8.

Recirculation flow evolution with lambda value at 9.55.

Figure 9.

Recirculation flow evolution with lambda value at 19.75.

Figure 10.

Recirculation flow bifurcation diagram. Continuous lines represent the values of stable solutions (feasible), and discontinuous lines represent unstable solutions (non-feasible).

The data in Table 2 are used to create a set of points on plane λ-τ, meaning points (λ,τ+) and (λ,τ-) are graphically represented for some of the analyzed cases (Table 3). The result is seen in the graph of Figure 10. An analysis of the graph in Figure 10 reveals the evident existence of bifurcation on point (0,0), which corresponds to a double point bifurcation of the pitchfork-type (supercritical according to Figure 1).

More thorough analysis shows that there are two consecutive bifurcations and two bistable states. The first bistable state is limited to lambda values ranging between 0 and 15 (pitchfork bifurcation), whereas the second bistable state occurs when values are larger than 15 (double point bifurcation).

4.3 Analysis of results (phase 3)

In Figure 9 bifurcation diagram is created to justify the behavior of recirculation loop B in a BWR6 unit, which is the system under analysis. As shown in Eq. (8), bifurcation point slopes on stability curves are determined by Eqs. (19) and (20).

$$\frac{{\mathrm{dq}}^{1)}}{\mathrm{dA}}={\mathrm{q}}_{\mathrm{A}}^1=\frac{{\mathrm{\tau }}^{+}\left({\mathrm{\lambda }}_2\right)-{\mathrm{\tau }}^{+}\left({\mathrm{\lambda }}_1\right)}{{\mathrm{\lambda }}_2-{\mathrm{\lambda }}_1}=\frac{{\mathrm{\tau }}^{+}\left(1.45\right)-{\mathrm{\tau }}^{+}\left(0\right)}{1.45-0}=\frac{0.293}{1.45}=0.2020$$E19

$$\frac{{\mathrm{dq}}^{2)}}{\mathrm{dA}}={\mathrm{q}}_{\mathrm{A}}^2=\frac{{\mathrm{\tau }}^{-}\left({\mathrm{\lambda }}_2\right)-{\mathrm{\tau }}^{-}\left({\mathrm{\lambda }}_1\right)}{{\mathrm{\lambda }}_2-{\mathrm{\lambda }}_1}=\frac{{\mathrm{\tau }}^{-}\left(1.45\right)-{\mathrm{\tau }}^{-}\left(0\right)}{1.45-0}=\frac{-0.674}{1.45}=-0.4648$$E20

Calculations are made using the data in Table 2.

The bifurcation diagram curve leads to believe that there is another possible bifurcation (double point), which is not of the pitchfork type. The bifurcation observed when the FCV point equals 77% has two slopes, feature of a double point:

The slope of the upper leg is determined by Eqs. (21) and (22).

$$\frac{{\mathrm{dq}}^{1)}}{\mathrm{dA}}={\mathrm{q}}_{\mathrm{A}}^1=\frac{{\mathrm{\tau }}^{+}\left({\mathrm{\lambda }}_2\right)-{\mathrm{\tau }}^{+}\left({\mathrm{\lambda }}_1\right)}{{\mathrm{\lambda }}_2-{\mathrm{\lambda }}_1}=\frac{{\mathrm{\tau }}^{+}\left(19.75\right)-{\mathrm{\tau }}^{+}\left(15.25\right)}{19.75-15.25}=\frac{1.013}{4.5}=0.2251$$E21

$$\frac{{\mathrm{dq}}^{2)}}{\mathrm{dA}}={\mathrm{q}}_{\mathrm{A}}^2=\frac{{\mathrm{\tau }}^{+}\left({\mathrm{\lambda }}_2\right)-{\mathrm{\tau }}^{+}\left({\mathrm{\lambda }}_1\right)}{{\mathrm{\lambda }}_2-{\mathrm{\lambda }}_1}=\frac{{\mathrm{\tau }}^{+}\left(15.25\right)-{\mathrm{\tau }}^{+}\left(9.55\right)}{15.25-9.55}=0.0068\left(\text{unstable}\right)$$E22

The slope of the lower leg is determined by Eqs. (23) and (24).

$$\frac{{\mathrm{dq}}^{1)}}{\mathrm{dA}}={\mathrm{q}}_{\mathrm{A}}^1=\frac{{\mathrm{\tau }}^{-}\left({\mathrm{\lambda }}_2\right)-{\mathrm{\tau }}^{-}\left({\mathrm{\lambda }}_1\right)}{{\mathrm{\lambda }}_2-{\mathrm{\lambda }}_1}=\frac{{\mathrm{\tau }}^{-}\left(19.75\right)-{\mathrm{\tau }}^{-}\left(15.25\right)}{19.75-15.25}=\frac{0.955}{4.5}=0.2122$$E23

$$\frac{{\mathrm{dq}}^{2)}}{\mathrm{dA}}={\mathrm{q}}_{\mathrm{A}}^2=\frac{{\mathrm{\tau }}^{+}\left({\mathrm{\lambda }}_2\right)-{\mathrm{\tau }}^{+}\left({\mathrm{\lambda }}_1\right)}{{\mathrm{\lambda }}_2-{\mathrm{\lambda }}_1}=\frac{{\mathrm{\tau }}^{+}\left(15.25\right)-{\mathrm{\tau }}^{+}\left(9.55\right)}{15.25-9.55}=0.0204$$E24

The two new curved stability slopes are nearly parallel, meaning stable states are permanent and the bistable state is not eliminated until 82% values (recorded at the plant) are reached. Thus, bifurcation is permanent and stable.

The existence of bistable flow is proven as a physical reality of the mathematical bifurcation concept. However, it is necessary to relate flow states with mathematical states and their physical meaning. This interpretation is based on the results of this work and those of the hydraulic model [10, 11] and computational fluid flow models (CFD) [13].

The recirculation flow rate can have three stable states, grouped in twos. The first two states, corresponding to FCV opening values between 62.3% and 77%, are defined by the vortex dynamics of flow adaption in the so-called cross-piece. These states cause the following effects:

• Vortex formation on the side legs of the manifold, in normal operation.

• The newly formed vortexes cause flow to take two forms: one for direct inlet at high flow and the other for helical current at low flow (Figure 11).

Figure 11.

Recirculation loop cross-piece flow conditions for first bifurcation.

In the first flow shape, pressure loss is as designed, and flow is quite similar to its theoretical value. In the second flow shape, pressure loss is higher since helical flow has more internal fluid friction and pipe wall friction, causing flow to be lower.

The next bistable scenario occurs when FVC opening is 77%. In this case, flow equals rated values and some are slightly higher (1%). In this state, flow takes two shapes: one of direct inlet at high flow (coinciding with the former high flow value) and the other free of central connection vortexes (which are dragged by the high flow); hence reducing internal friction and enhancing flow (Figure 12).

Figure 12.

Recirculation loop cross-piece flow conditions for second bifurcation.

5. Conclusions

The bistable flow state is the result of an evolved dynamic system which is based on a specific FCV opening value, offers two possible states. Transitioning from one to the other is determined by system evolution in all possible values and their flow rate variance.

Graphically, in each moment, the system has a balance point determined by a minimal energy state represented by an attraction basin, causing the system to evolve in it and be attracted to the minimum point, depending on the noise (turbulence). As FCV opening value increases, flow and turbulence also increase, causing the attraction basin to be increasingly larger although the minimum point is preserved. There comes a moment when the attraction basin begins splitting in two, growing in the middle, between both parts, a peak area that separates them. When the bistable flow is fully formed, both attraction basins and their separation are obvious. The system moves between these two “balance” values (Figure 13).

Figure 13.

Energy or attraction basins and separation peak.

The point where the separation appears and both attraction basins begin to take shape is unique and marks the start of bifurcation. This separation coincides with the unstable system solution, meaning that if the system has a solution for such separation, it will evolve towards one of the two values provided by the attraction basin.

With regards to value changes and fluctuations, having two energy pits lead to some changes between them due to the intermediate peak value being exceeded. This is possible because the turbulence generated at high numbers of Reynolds increases the fluid energy locally, which exceeds the intermediate peak and causes the system to enter the other energy state. This theory is coherent with the characterization of bistable flow as a noise-induced transition phenomenon, considering that in this case, noise is the actual flow turbulence (Figure 14).

Figure 14.

System energy function and correlation with bistable flow.

To conclude, the Navier-Stokes equation bifurcation theory for the recirculation system explains all events reported, all simulations made, and all effects observed. Likewise, this theory provides the grounds to justify that only actions intended to eliminate turbulence formation and its detachment from the cross-piece are adequate to remove bistable flow. Thus, cross-piece modification alters the system curve (flow-opening) causing the unique point of double tangent to move to another location and turn currently bifurcated points into regular points. This modification from unique point to regular point causes flow to stop being bistable.

Lastly, it is worth mentioning that this study focuses on a fixed, complex geometry system, proves that turbulence and vorticity evolution are not a continuous function in relation to flow or the number of Reynolds. On the contrary, it has a bifurcation area turbulence that is discontinuous while drifting and has a continuous, non-derivable function.