Numerical simulation of the shock wave / boundary layer interaction in a shock tube by using a high resolution monotonicity-preserving scheme. V. Daru1 and C. Tenaud2 1 2
Laboratoire SINUMEF, ENSAM, 151 Boulevard de l’Hˆ opital 75013 Paris, FRANCE
[email protected] LIMSI - UPR CNRS 3251, BP.133, 91403 ORSAY Cedex, FRANCE.
[email protected]
Summary. This work concerns the flow generated in a shock tube after the shock wave has reflected at the end wall. When the fluid is viscous, a complex unsteady interaction takes place between the reflected shock wave and the incident boundary layer. The numerical simulation of this problem necessitates numerical schemes which are robust and very accurate. Thanks to the use of a one step high order accurate scheme that we recently developed [2], we have obtained converged results for several values of the Reynolds number. We then analyze in details the interaction mechanisms and the flow dynamics.
1 Introduction The shock tube problem may be found in various real applications such as, for instance, in high speed aerodynamic flow facilities either to study the chemical relaxation process in high temperature gas mixture or to generate high enthalpy reservoir conditions. In such a configuration, one wishes ideally to get a uniform state downstream of the reflected shock wave near the end wall. In fact, the reflected shock wave interacts with the incident boundary layer. This process modifies however the gas properties in this end wall region. A complex unsteady interaction develops between the reflected shock wave and the boundary layer. When the stagnation pressure within the boundary layer is lower than the one downstream of the normal reflected shock wave, the boundary layer separates and a lambda-like shock wave pattern occurs. The numerical simulation of this problem necessitates numerical schemes which are both robust and very accurate, meaning those which can represent with high accuracy both the smooth regions and capture discontinuities with the robustness that is common to Godunov-type methods. In a previous study [1], we exhibited the convergence problems encountered by using classical shock capturing TVD approaches on such a test-case. We recently developed an original high order coupled time-space scheme (named OSMP7, accurate up to the 7th-order) which guaranties Monotonicity Preserving properties [2, 3]. The OSMP7 scheme gives very accurate results on numerous classical test cases [2, 3]. Compared to classical WENO schemes, the OSMP7 schemes better capture the discontinuities with a comparable order of accuracy in the regular regions. However, the OSMP7 schemes are sixth fold lower CPU cost than the WENO schemes.
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The high accurate OSMP7 scheme was applied to the shock tube problem. Converged results were obtained and are presented for four Reynolds numbers: Re = 200, 500, 750 and 1000, with a grid independency assessment and compared with the results published by Sj¨ ogreen et al. [5]. The present results allow us to analyze the dynamics of the shock wave / boundary layer interaction at the end wall of the shock tube.
2 Numerical procedure The non-dimensionalized Navier Stokes equations are written, using the cartesian coordinates, as follows: wt + (f (w) − f v (w, wx , wy ))x + (g(w) − g v (w, wx , wy ))y = 0
(1)
where w = (ρ, ρu, ρv, ρE)t is the vector of the conservative variables using the classical notations. In addition, a perfect gas law is used, assuming a constant specific heat ratio; in the following, the calculations are performed by using γ = 1.4 (for air). The vectors named f , g et f v et g v are, respectively, the Euler and viscous fluxes in the two space directions (x and y, respectively). The discretization of the Euler fluxes is performed by using the recently developed OSMP7 scheme [3]. This scheme has been developed on a coupled time and space Lax-Wendroff approach. The OSMP7 scheme is accurate up to the seventh order in time and space, at least in the scalar (linear and non-linear) case. For shock capturing purpose, a Monotonicity-Preserving (MP) constraint is incorporated to the scheme which is demonstrated to be equivalent to the TVD condition [3], except in the vicinity of the extrema where the TVD constraints are relaxed to get a more accurate solution. In this way, this scheme allows both to capture sharp discontinuities without producing spurious oscillations in their vicinity and to preserve high order of accuracy in the smooth regions. The viscous fluxes are discretized through the use of a classical second-order centered scheme. In the multidimensional case, a Strang splitting is used. For more details about the numerical procedure, see references [2, 3].
3 Dynamic flow analysis of a shock wave / boundary layer interaction. We consider a unit side length square shock tube with insulated walls. The diaphragm is initially located in the middle of the tube (x = 0.5). The initial state, in terms of dimensionless quantities, is on the left of the diaphragm: ρL = 120, pL = ρL /γ, uL = vL = 0, and on the right: ρR = 1.2, pR = ρR /γ, uR = vR = 0. At the initial time, the diaphragm is broken. The inviscid solution is represented in Fig. 1, showing the evolution of the density in the x − t plane along a line situated at the middle hight of the tube. A shock wave, followed by a contact discontinuity, moves to the right (the shock Mach number is equal to 2.37). The incident shock wave is weak, and reflects at the right end wall approximately at time t = 0.2. After
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this reflection, it interacts with the contact discontinuity. Then, complex interactions occur. The contact discontinuity stays stationary, close to the right end wall. Afterwards, the reflected shock wave begins to interact with the rarefaction wave at time t = 0.4. On the other side, the rarefaction waves reflect on the left end wall, at a dimensionless time t = 0.5, modifying the propagation of the incident rarefaction wave. In the viscous case, the incident shock wave and contact discontinuity, during their
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propagation, interact with the horizontal wall, creating a thin boundary layer. After its reflection on the right end wall, the shock wave interacts with this boundary layer. As the stagnation pressure in the boundary layer is lower than the one within the outflow region, a dead-water region (named “bubble”) takes place over a large extent within the boundary layer, resulting in a major modification of the flow pattern and the formation of “a lambda-shape like shock pattern”. This mechanism is illustrated in Fig. 2, showing the density contours at t = 0.6 for a Reynolds number flow Re = 200. The triple point emerging from the lambda-shape like shock pattern generates a slip line which rolls up around the vortex situated at the bottom of the contact discontinuity (bottom right corner). Though the dynamics of the flow beneath the lambda-shape like shock pattern seems very complex, we try however to analyze and understand its time evolution which is reported in figure 2, for a Reynolds number flow Re = 200 on the density contours at four dimensionless times (t = 0, 4 ; t = 0, 5 ; t = 0, 6 ; t = 0, 7). From t = 0, 4, the main stream, flowing in between the lambda-shock and the bubble, rolls up around the dead-water region by forming a high-speed jet which impacts the bottom wall (Fig 2 top-left). At t = 0.5, this jet, deviated up by the solid wall, then impacts the supersonic shear layer (slip line) coming from the separation point (Fig 2 bottom-left). Hence, the jet is one more time, deviated down toward the
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bottom wall. As time goes on, these reflection mechanisms alternatively between the bottom wall and the slip line, has been carried on (t = 0, 6 Fig 2 top-right and t = 0, 7 Fig 2 bottom-right). This counter jet flow is very energetic since it can reach supersonic speeds at several places as we can see on the Mach number contours (Fig 3-left). On this figure, to better exhibit the supersonic jet path, we superposed the streamline traces on the iso-contours. Each impact of the jet, either on the solid wall or on the slip line, corresponds to a relatively high pressure region (Fig 3-right) and in between, relatively low pressure regions are created. Hence, the flow rolls up around these relatively low pressure zones to create counter-rotative secondary vortices. Let us mention that the dynamical process exhibited here to explain the formation of the counter-rotative vortices within the separated region, has already been observed in the flow around a circular cylinder when the boundary layer separates [7]. At each time the jet flow impacts the slip line coming from the separation point, this slip line is deformed (Fig 3). As the flow passing in between the lambda-shape like shock pattern and the separation is supersonic, the wrapping of the slip line delimiting the main separation region, induces shocklets which interact with the main shock pattern to produce small Mach stems, as we can see just under the triple point (Fig 3).
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Fig. 2. Density contours for Re = 200 at several dimensionless time (from top to bottom and left to right): t = 0, 4, t = 0, 5, t = 0, 6, t = 0, 7.
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Fig. 3. Mach number (left) and static pressure (right) contours for a Reynolds number Re = 500 at a dimensionless time t = 0, 55.
4 Influence of the Reynolds number. 4.1 Grid convergence study. The simulations have been carried out for four Reynolds numbers: Re = 200, Re = 500, Re = 750 and Re = 1000. Several meshes with increasing resolutions have been used to guaranty solution grid independencies. Let us notice that all the meshes employed use equally spaced points with cell aspect ratio equal to unity. The grid convergence is demonstrated for instance on the density distribution along the bottom wall for two Reynolds numbers: Re = 500 and Re = 750 (Fig 4). As it was expected, the greater the Reynolds number, the finer the grid. While the results do not fit exactly at the highest resolution, the differences remain very small and we can consider that, for the following Reynolds numbers, the results are converged when the grid resolutions are: 1000x500 for Re = 200, 1500x750 for Re = 500 and 2000x1000 for Re = 750. For Re = 1000, the discrepancies between the results obtained on the following grid resolutions 2000x1000 and 3000x1500 are significative and thus the convergence could not be considered as completely reach. Calculations on finer grids could not be carried out because of the prohibited CPU time required. To validate the present results and consider them as reference results, we performed calculations on the same configurations with a more classical scheme, based on a third order Runge-Kutta time integration and a WENO (r=5) scheme [4] as spatial discretization. Using this classical approach, results have been obtained very close to the OSMP7 ones. Nevertheless, the grid convergence cannot be reach in several grid resolutions using the WENO scheme because of the prohibited CPU time required which is at least six times greater than the OSMP7 scheme. Moreover concerning the convergence process, the OSMP7 scheme presents a more rapid grid convergence than the WENO scheme.
4.2 Reynolds number influence on the flow dynamics. The analysis of the counter jet flow dynamics within the separation region have pointed out that, increasing the Reynolds number, the speed of the counter flow jet grows and therefore, during a fixed period, the number of the jet alternated reflections between the bottom wall and the slip line increases. Consequently, the number of counter rotative vortices produced in this interaction region might increase too.
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For the lowest Reynolds numbers studied Re = 200 and Re = 500, this dynamical process has been recorded and is visible on Fig 5 where the vorticity contours are plotted for four Re numbers at a dimensionless time t = 1. Nevertheless, this process does not persist from Re = 750 (Fig 5) since the regular arrangement of the largest eddy structures seems to disappear. During the set up of the large scale eddy structures (in blue color on Fig 6), eddy structures with positive vorticity (in red on Fig 6) are created close to the bottom wall (around x = 0, 7, for instance), with an increasing enstrophy with the Reynolds number. In a similar manner as when a vortex interacts with a boundary layer, when the enstrophy of these eddy structures is high enough, they are repulsed out of the boundary layer. Hence, the structures are stretched and distorted as an intense vortex filament. Due to the shearing effect of the outflow, this filament is elongated up to the bursting of the filament. When this filament is broken up, the enstrophy intensity decreases close to the wall which results in the retraction of the structure with a “spring back effect”. The two consecutive structures with a negative vorticity (in blue color) then merge. This dynamical process described herein is carried out with strengthening, up to a pair of counter-rotative vortices could be burst out of the boundary layer.
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Fig. 4. Density distributions along the bottom wall of the shock tube, obtained on several fine grids for Re = 500 (left) and Re = 750 (right).
5 Conclusion The present study exhibits that the calculation of such a complex shock tube problem necessitates numerical schemes which are both robust and very accurate. The OSMP7 scheme allows us to reach fully converged solutions for several Reynolds numbers which could not be reached by using the classical WENO scheme because of the too large CPU cost. The present shock tube problem can be considered as a good test-case for checking high-resolution schemes. The present solutions we obtained in this study with the OSMP7 scheme could be considered as reference solutions on such a configuration.
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Fig. 5. Vorticity contours at a dimensionless time t = 1, obtained for four Reynolds numbers; from top to bottom and left to right: Re = 200 ; Re = 500 ; Re = 750 ; Re = 1000. rot: -250-230-210-190-170-150-130-110 -90 -70 -50 -30 -10 10 30 50 70 90 110 130 150 170 190 210 230 250
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Fig. 6. Vorticity contours obtained for Re = 750 at several dimensionless time; from top to bottom and left to right: t = 0, 80 ; t = 0, 85 ; t = 0, 90 ; t = 0, 95.
References 1. V. Daru & C. Tenaud 2001 ‘Evaluation of TVD high resolution schemes for
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V. Daru and C. Tenaud unsteady viscous shocked flows’, Computers and Fluids, 30, 89–113. 2. V. Daru & C. Tenaud 2002 ‘High resolution monotonicity-preserving schemes for unsteady compressible flows’, International Conference on Conputational Fluid Dynamics, July 2002, Sydney, Australia. 3. V. Daru & C. Tenaud 2004 ‘High order one-step monotonicity-preserving schemes for unsteady compressible flow calculations’, Journal of Computational Physics, 193, 563–594. 4. G.S. Jiang & C.W. Shu 1989 ‘Efficient implementation of weighted ENO schemes’, Journal of Computational Physics, 83, 32–79. 5. B. Sjogreen, H.C. Yee: ‘Grid convergence of high order methods for multiscale complex unsteady viscous compressible flows’, Journal of Computational Physics, 185, 1–26 (2003). 6. A. Suresh & H.T. Huynh 1997 ‘Accurate Monotonicity-Preserving Schemes with Runge-Kutta Time Stepping’, Journal of Computational Physics, 136, 83–99. 7. Ta Phuoc Loc & R. Bouard 1985 ‘Numerical solution of the early stage of the unsteady viscous flow around a circular cylinder : a comparison with experimental visualization and measurements’, Journal of Fluid Mechanics, 160, 93–117.
