Mizushima, Jiro

Tanaka, Hidekazu

Kyoto University Research Information Repository

TitleSymmetry-Breaking Bifurcation of Radially Outgoing Flowbetween Two Disks (Generation-Sustenance Mechanism andStatistical Law of Turbulence)Author(s)Mizushima, Jiro; Tanaka, HidekazuCitation数理解析研究所講究録 (2002), 1285: 122-129Issue Date2002-09URL http://hdl.handle.net/2433/42437RightType Departmental Bulletin PaperTextversionpublisherKyoto UniversitySymmetry-Breaking Bifurcation of Radially Outgoing Flowbetween Two Disks(Jiro MIZUSHIMA) (Hidekazu TANAKA)Department of Mechanical Engineering,Doshisha University1. IntroductionRadially outgoing flow between two parallel circular disks is asimple model for flows in the injectionmolding of plastic, hydrostatic air bearings and centrifugal compressor diffusers. The local Reynoldsnumber defined by using the half distance between two circular disks and the local representative velocitydecreases as the flow goes downstream between two disks.Appearance of separation vortices in the flow field was reported by $\mathrm{I}\mathrm{s}\mathrm{h}\mathrm{i}\mathrm{z}\mathrm{a}\mathrm{w}\mathrm{a}^{1,2)}$ , that is explained bythe presence of an adverse pressure gradient at small radii. He analyzed the flow by applying the boundarylayer theory with aseries-expansion method for the inlet region and with amomentum-integral method forthe downstream region on the assumption of symmetric flow field, and predicted the threshold Reynoldsnumber $Re_{\mathrm{t}}$ for the appearance of the separation vortices as $Re_{\mathrm{t}}\sim 100$ . It is added that Ishizawa assumedauniform flow at the inlet section and the non-dimensional inlet radius, the ratio of the inlet radius tothe half width between two disks, being unity. The separation vortices were also found to appear in thenumerical results by $\mathrm{R}\mathrm{a}\mathrm{a}1^{3)}$ who made numerical calculations of the steady-state solution by using finitedifference approximation on the assumption of symmetric flow field along the center line between twodisks for the same flow field as Ishizawa treated. However, the value of the throshold Reynolds numberevaluated by Raal is 60 which differs significantly from the value $Re_{\mathrm{t}}\sim 100$ predicted by Ishizawa.Mochizuki and $\mathrm{Y}\mathrm{a}\mathrm{n}\mathrm{g}^{4)}$ investigated the instability of the flow numerically and experimentally. Theflow-visualization method was employed in their experiments and the dynamical vorticity transport equa-tions were solved by using finite difference approximation without assuming the symmetry along thecenterline between two disks in numerical simulations. They observed oscillatory flows above acriticalReynolds number, and evaluated the value of the critical Reynolds number numerically and experi-mentally, where agood accordance between numerical and experimental evaluations was shown. Thenon-dimensional inlet radius they adopted is 13.3 which differs from the value of Ishizawa or Raal. So,the direct comparison of the results obtained by Mochizuki and Yang with those by Ishizawa and Raalis difficult. However it is concluded that the flow field is symmetric below the critical Reynolds numberand the flow becomes oscillatory, which inevitably accompanies asymmetric flow fields.Instability of radial outgoing flow between two parallel circular disks and its transition is investigatedby numerical simulations and the linear stability analysis in the present paper. We adopt the sameconfiguration of the problem with $\mathrm{I}\mathrm{s}\mathrm{h}\mathrm{i}\mathrm{z}\mathrm{a}\mathrm{w}\mathrm{a}^{1,2)}$ and $\mathrm{R}\mathrm{a}\mathrm{a}\mathrm{l}.3$ ) The effect of the value of the non-dimensionalinlet radius on the transition flow is also considered. Axisymmetric incompressible flow field is assumed,but the symmetry along the centerline between two disks is not assumed. An analysis of the asymptoticsolution in the far field from the center is also given1285 2002 122-1291222. FormulationConsider aradially outgoing flow between two parallel circular disks with the spacing $2h^{*}$ and theinfinite outer radius (Fig. 1). Flow enters into the gap between the two disks from the inlet AB at the(a) (b)Fig. 1Configuration and coordinates.inner radius $r_{1}^{*}$ with auniform velocity profile. We define two non-dimensional parameters, i.e., theReynolds number and the non-dimensional inlet radius, as$Re \equiv\frac{Q^{*}}{4\pi\nu h^{*}}$ , $r_{1}\overline{=}r_{1}^{*}/h^{*}$ , (1)where $Q^{*}$ is the volumetric flow rate through the gap between two disks and $\nu$ is the kinematic viscosityof the fluid.2.1 Fundamental equationWe introduce the stream function $\psi(r, z, t)$ for the axisymmetric flow in the cylindrical coordinates.The governing equations for the vorticity component $\omega(r, z, t)$ in the circumferential direction and thestream function $\psi(r, z, t)$ are written in anon-dimensional form as$\frac{\partial\omega}{\partial t}-\frac{1}{r}\frac{\partial(\psi,\omega)}{\partial(r,z)}-\frac{1}{r^{2}}\frac{\partial\psi}{\partial z}\omega$ $= \frac{1}{Re}\Delta\omega$ , (2)$\omega=\frac{1}{r}D^{2}\psi$ , (3)where$\frac{\partial(f,g)}{\partial(r,z)}\equiv\frac{\partial f}{\partial r}\frac{\partial g}{\partial z}-\frac{\partial f}{\partial z}\frac{\partial g}{\partial r}$ , $\Delta\equiv\frac{\partial^{2}}{\partial r^{2}}+\frac{1}{r}\frac{\partial}{\partial r}+\frac{\partial^{2}}{\partial z^{2}}-\frac{1}{r^{2}}$ , $D^{2} \equiv\frac{\partial^{2}}{\partial r^{2}}-\frac{1}{r}\frac{\partial}{\partial r}+\frac{\partial^{2}}{\partial z^{2}}$,and all the physical variables are made non-dimensional by using the representative length $h^{*}$ and therepresentative velocity $Q^{*}/4\pi h^{*2}$ .The flow is assumed to enter the inlet (AB in Fig. 1) with auniform velocity profile so that theboundary condition is written as$\psi=z$ , $\omega=0$ , $(r=r_{1})$ . (4)The boundary condition at the two disks is the nonslip boundary condition that is expressed by$\psi=\pm 1$ , $\omega=\frac{1}{r}\frac{\partial^{2}\psi}{\partial z^{2}}$ , $(z=\pm 1)$ , (5)123where the complex signs are taken in the same order. The flow field is assumed to extend to infinitelylarge radius, therefore the flow field has the asymptotic velocity at sufficiently large distance from thecenter, which is derived in the next subsection. Then the boundary condition at the end of computationaldomain (CD in Fig. 1) is expressed as$\psi=\frac{1}{2}(3z-z^{3})+\frac{3}{8Re}(\frac{1}{7}z-\frac{11}{35}z^{3}+\frac{1}{5}z^{5}-\frac{1}{35}z^{7})\frac{1}{r^{2}}$ ,$\omega$ $= \frac{3}{Re}(\frac{1}{7}z-\frac{11}{35}z^{3}+\frac{1}{5}z^{5}-\frac{1}{35}z^{7})\frac{1}{r^{5}}-3\frac{z}{r}+\frac{3}{4Re}(-\frac{33}{35}z+2z^{3}-\frac{3}{5}z^{5})\frac{1}{r^{3}}$,$(r=r_{2})$ . (6)2.2 Asymptotic solution for far fieldIt is expected that the outgoing flow between two disks approaches to the fully developed planePoiseuille flow far downstream. We make an asymptotic analysis to derive the profile of the flow field atfar distance. Anew variable defined by$\xi=\frac{1}{r}$ , (7)is introduced, then the flow region is transformed from $r$ $=[r_{1}, \infty)$ to $\xi=[0,1/r_{1}]$ . Steady flow field isassumed at very far distance, therefore Eqs. (2) and (3) are rewritten as$\xi^{3}\frac{\partial\psi}{\partial\xi}\frac{\partial}{\partial z}\Lambda\psi-\xi^{3}\frac{\partial}{\partial\xi}\Lambda\psi-\xi^{2}\frac{\partial\psi}{\partial z}\Lambda\psi=\frac{1}{Re}\Lambda^{2}\psi$, (8)at far downstream, where$\Lambda=\xi^{4}\frac{\partial^{2}}{\partial\xi^{2}}+3\xi^{2}\frac{\partial}{\partial\xi}+\frac{\partial^{2}}{\partial z^{2}}$ .We explore the solution $\psi(\xi, z)$ of Eq. (8) in aseries of expansion in $\xi(=1/r)$ expressed as$\psi(\xi, z)=\psi_{0}+\xi\psi_{1}+\xi^{2}\psi_{2}+O(\xi^{3})$ , (9)where only the terms up to $O(\xi^{2})$ are retained. The boundary conditions for the expansion coefficients$\psi_{n}$ are written as$\psi_{0}=\pm 1$ , $\frac{\partial\psi_{0}}{\partial z}=0$ , $(z=\pm 1)$ ,$\psi_{n}=0$ , $\frac{\partial\psi_{n}}{\partial z}=0$ , $(z=\pm 1)$ , (10)where the complex signs should be taken in the same order and $n$ should taken to be 1or 2. Bysubstituting the expansion (9) into Eq. (8) and equating the terms with the same power of 4, we obtainthe equations for $\psi_{n}$ , $(n =0,1,2)$ . After solving the resultant equation for $\psi_{n}$ , we obtain the asymptoticexpression of $\psi$ at large distance as expressed as$\psi(r, z)=\frac{1}{2}(3z-z^{3})+\frac{3}{8Re}(\frac{1}{7}z-\frac{11}{35}z^{3}+\frac{1}{5}z^{5}-\frac{1}{35}z^{7})\frac{1}{r^{2}}$. (11)where the solutions are expressed in $r$ by substituting $r$ $=1/\xi$ . The asymptotic solution (11) is adoptedas the boundary condition for our numerical simulations and the numerical calculations of the steady andsymmetric solutions1242.3 Steady and symmetric solutionSteady and symmetric solution along the centerline between two disks exists regardless of the mag-nitude of the Reynolds number, which is unstable at the Reynolds numbers above acritical value. Thesteady and symmetric solution $(\overline{\psi},\overline{\omega})$ satisfies the steady-state equations, which are obtained by droppingthe terms with time derivative $\partial/\partial t$ in Eq. (2).The steady and symmetric solution can be calculated in the upper or lower half domain of ABCDin Fig. 1therefore the boundary condition on the centerline between the two disks is expressed as$\overline{\psi}=0$ , $\overline{\omega}=0$ , $(z=0)$ . (12)The boundary conditions for $(\overline{\psi},\overline{\omega})$ at the inlet $r=r_{1}$ , the outlet $r=r_{2}$ and the top or the bottom disksare the same as that of Eqs. (2) and (3).2.4 Disturbance equationWe consider adisturbance $(\psi’, \omega’)$ added to the steady and symmetric solution $(\overline{\psi},\overline{\omega})$ and expressthe stream function and the vorticity $(\psi, \omega)$ as $\psi=\overline{\psi}+\psi’$ and $\omega$ $=\overline{\omega}+\omega’$ to investigate the stabilityof the steady and symmetric solution. The linear disturbance equations are obtained by substitutingthe expressions for $\psi=\overline{\psi}+\psi’$ and $\omega$ $=\overline{\omega}+\omega’$ into Eqs. (2) and (3), subtracting the steady-stateequations, and neglecting the higher order terms than the linear terms concerning the disturbance $\psi’$and $\omega’$ . Furthermore, the solution $(\psi’, \omega’)$ can be shown to have the form of $\psi’=\hat{\psi}\exp(\lambda t)$ , and$\omega’=\hat{\omega}\exp(\lambda t)$ , then substitution of these expression into $(\psi’, \omega’)$ leads the equations for $(\hat{\psi},\hat{\omega})$ as$\lambda\hat{\omega}-\frac{1}{r}\{\frac{\partial\overline{\psi}}{\partial r}\frac{\partial\hat{\omega}}{\partial z}+\frac{\partial\hat{\psi}}{\partial r}\frac{\partial\overline{\omega}}{\partial z}-\frac{\partial\overline{\psi}}{\partial z}\frac{\partial\hat{\omega}}{\partial r}-\frac{\partial\hat{\psi}}{\partial z}\frac{\partial\overline{\omega}}{\partial r}\}-\frac{1}{r^{2}}\{\frac{\partial\overline{\psi}}{\partial z}\hat{\omega}+\frac{\partial\hat{\psi}}{\partial z}\overline{\omega}\}=\frac{1}{Re}\Delta\hat{\omega}$, (13)$\hat{\omega}=\frac{1}{r}D^{2}\hat{\psi}$ . (14)The boundary conditions for $(\hat{\psi},\hat{\omega})$ are written as$\hat{\psi}=0$ , $\hat{\omega}=0$ , $(r=r_{1}, r_{2})$ ,$\hat{\psi}=0$ , $\hat{\omega}=\frac{1}{r}\frac{\partial^{2}\hat{\psi}}{\partial z^{2}}$ , $(z=\pm 1)$ . (15)The disturbance equation can be soIved in the upper or lower half domain of ABCD in Fig. 1thereforethe boundary condition on the centerline between the two disks is expressed as$\hat{\psi}(r, z)=\hat{\psi}(r, -z)$ , $\hat{\omega}(r, z)=\hat{\omega}(r, -z)$ , $(z=0)$ . (16)3. Numerical method3.1 Numerical simulationIn numerical simulations by the time marching method, an equally spaced mesh system with $\Delta r=$$\Delta z=5\cross 10^{-2}$ is used. The vorticity transport equation is solved by the Runge-Kutta method withthe fourth-0rder accuracy in time together with the second-0rder accuracy of central finite difference inspace. The time increment At is chosen as $\Delta t=1\cross 10^{-4}$ . The Poisson equation is discretized by thesecond-0rder central finite difference and solved by the SOR method, where the relaxation factor $\epsilon$ istaken as $\epsilon=1.82$1253.2 Steady and symmetric solution and linear stability analysisBoth the steady-state vorticity transport equation and the Poisson equation are solved by the SORiterative method to calculate steady and symmetric solutions. Spatial derivatives are approximated by thesecond-0rder finite differences. The relaxation factor $\epsilon$ for the SOR method is determined by consideringthe aspect ratio and the Reynolds number in the range $0.1<\epsilon<1.0$ . In order to calculate unstablesteady and symmetric solutions above acritical Reynolds number, the SOR method is utilized under thesymmetry condition along the centerline between two disks. This method is used also for the numericalevaluation of the linear growth rate.4. Numerical results4.1 Flow patterns and bifurcation(a) (b)Fig. 2Flow pattern. $Re=40$ , $r_{1}=1$ . (a) Streamlines, (b) Velocity profiles.For the case of the non-dimensional inlet radius $r_{1}$ $=1$ , we have done numerical simulations for theReynolds numbers in the range of $10\leq Re\leq 200$ . The outgoing flow is symmetric at small Reynoldsnumbers. Atypical example of such asteady and symmetric flow is shown in Fig. 2for $Re=40$ .Streamlines for the symmetric flow is depicted in Fig. 2(a) and the velocity profiles in the z-directionis shown in Fig. 2(b). The streamlines gather together just downstream of the inlet as seen in Fig. 2(a), where jet-like flow is observed in Fig. 2(b). Another flow pattern shown in Fig. 3 $(Re=50)$ isalso asteady and symmetric flow, but differs from that in Fig. 2in presence of separation vortices. Theseparation vortices in Fig. 3(a) are formed lying in $r$ $=1.85-3.55$ near both the upper and lower walls.(a) (b)Fig. 3Flow pattern. $Re=50$ , $r_{1}=1$ . (a) Streamlines, (b) Velocity profiles.The flow pattern obtained in numerical simulations at $Re=70$ is steady, but asymmetric as depictedin Fig. 4. The stream coming from the inlet is seen to bend on the lower wall. We can easily imagineanother asymmetric flow, which bends on the upper wall from the symmetry consideration of the system.Such asteady and asymmetric flow has not been reported before. The separation-vortex region in Fig. 4126(a) (b)Fig. 4Flow pattern. $Re=70$ , $r_{1}=1$ . (a) Streamlines, (b) Velocity profiles.(a) is larger than that in Fig. 3(a), and the separation vortex lies in $r=1.5-6.3$ near the upper wall.Fig. 5Bifurcation diagram, $r_{1}=1$ .We have confirmed by numerical calculations of the steady-state equations that the symmetric flowalso does exist at $Re=70$ . The asymmetric flow is thought to appear due to the instability of thesymmetric flow. In order to investigate the bifurcation of steady-state solution, we obtain the bifurcationdiagram by adopting arepresentation value $v_{1}$ , the velocity in $z$-direction at arepresentation point $\mathrm{P}_{1}$$[(r, z)=(2, \mathrm{O}),\mathrm{F}\mathrm{i}\mathrm{g}. 1]$ . The bifurcation diagram is depicted in Fig. 5. The bifurcation is determined asapitchfork bifurcation from the relation $v_{1}^{2}\propto$ $(Re -Re_{\mathrm{c}})$ , and the critical Reynolds number $Re_{\mathrm{c}}$ being62.7.4.2 Linear stability analysisWe investigate the linear stability of the steady and symmetric solution which is abasic solution ofthe outgoing flow for the case of the non-dimensional inlet radius $r_{1}=1$ . We have solved the steady-stateequations numerically to obtain the steady and symmetric solutions $(\overline{\psi},\overline{\omega})$ and also solved Eqs. (13) and(14) to evaluate the linear growth rate Afor the steady and symmetric solutions. Typical examples of asteady and symmetric flow and alinear disturbance are shown in Fig. 6for the Reynolds number $Re=64$ .Figure 6(a) shows the streamlines of the steady and symmetric solution, and the disturbance obtainedby solving Eqs. (13) and (14) is depicted in Fig. 6(b). The disturbance has significant magnitude in a127(a) (b)Fig. 6Flow pattern. $Re=64$, $r_{1}=1$ . Streamlines, (a) Unstable symmetric flow, (b) Disturbance,limited region near the inlet, which makes the symmetric flow to bend on one wall in that region.Fig. 7Linear growth rate $\lambda$ , $r_{1}$ $=1$ .The linear growth rate Ais evaluated for various Reynolds numbers, which is shown in Fig. 7for thenon-dimensional inlet radius $r_{1}=1$ . The critical Reynolds number $Re_{\mathrm{c}}$ is determined as $Re_{\mathrm{c}}=62.8$ fromFig. 7for the non-dimensional inlet radius $r_{1}=1$ . The relative error between the values of $Re_{\mathrm{c}}=62.7$obtained from the bifurcation diagram and $Re_{\mathrm{c}}=62.8$ from the linear stability analysis is about 0.16%,which shows the consistency between the two values.We have evaluated the critical Reynolds number $Re_{\mathrm{c}}$ for others values of the non-dimensional inletradus $r_{1}$ and depicted them against $r_{1}$ in Fig. 8in the range of $0.5\leq r_{1}$ $\leq 3$ . The flow is symmetric at theReynolds number under the solid line in Fig. 8, but becomes asymmetric at the Reynolds number abovethe line. The steady and asymmetric flow may make atransition into an oscillatory flow above anotherReynolds number, say $Re_{\mathrm{c}}’$ . However, it may be possible that the steady and symmetric flow makes atransition into an oscillatory flow without experiencing an asymmetric flow for the non-dimensional inletradii $r_{1}$ much larger than 3. The critical Reynolds number $Re_{\mathrm{c}}’$ may approach to 5771, where the planePoiseuille flow becomes unstable128Fig. 8Critical Reynolds number and $r_{1}$ : the non-dimensional inlet radus.5. ConclusionsWe made an asymptotic analysis for the flow profile at far distance and confirmed the flow fieldapproaches to the fully developed plane Poiseuille flow there. We have done numerical simulations,numerical calculations of the steady symmetric solutions and their linear stability analysis for the outgoingflow with finite difference approximations. As results, we found that the flow is symmetric at smallReynolds numbers, but becomes asymmetric above acritical Reynolds number. The transition into theasymmetric flow is determined due to apitchfork bifurcation judging from the relation $v_{1}^{2}\propto(Re-62.7)$for the non-dimensional inlet radius $r_{1}=1$ . The critical Reynolds number obtained from the linearstability analysis is 62.8 which is in good agreement with the critical value 62.7 evaluated by numericalsimulation data. The larger the value of the non-dimensional inlet radii $r_{1}$ is, the larger the value of thecritical Reynolds numbers $Re_{\mathrm{c}}$ for atransition into asymmetric flow becomes.References1) S. Ishizawa, Bulletin of JSME, 8, 353-367 (1965).2) S. Ishizawa, Bulletin of JSME, 9, 86-103 (1966).3) J. D. Raal, J. Fluid Mech., 85, 401-416 (1978).4) S. Mochizuki and W. J. Yang, J. Fluid Mech., 154, 377-397 (1985)129

Symmetry-Breaking Bifurcation of Radially Outgoing Flow between Two Disks (Generation-Sustenance Mechanism and Statistical Law of Turbulence)

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Symmetry-Breaking Bifurcation of Radially Outgoing Flow between Two Disks (Generation-Sustenance Mechanism and Statistical Law of Turbulence)

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