In this study, the hyperbolic method is adopted to explore the flow field states of incompressible flow in a four-sided lid-driven square cavity. In particular, we focus on the flow bifurcation obtained at the critical Reynolds number
The two-dimensional flow driven by boundaries moving inside a cavity is considered to be a classical problem of computational fluid mechanics. In this work, as in many studies, we consider a simple square cavity. This problem displays many interesting solutions, like vortex structures and multiple flow solutions [
The different studies have focused on cavity driven by a single plate, two parallel plates, two non-facing plates, or four-sided plates. In the latter configuration, the upper plate moves rightwards, the lower plate moves leftwards, the left plate moves downwards, and the right plate moves upwards. The four plates are also considered to move at the same velocity in the present work. A remarkable result is the existence of a bifurcation for a particular Reynolds number (Re) value close to R c = 130 . Indeed, the steady-state flow changes from a stable symmetric solution for a Reynolds number value lower than Rc to two stable asymmetric solutions for a higher Reynolds number. This is a pitchfork bifurcation as the symmetric solution becomes unstable beyond bifurcation. The value for Rc deduced from the different previous studies is not very well determined, as it varies between 129 and 130.4. A new evaluation is thus needed.
In the present study, we propose to revisit this problem by using the hyperbolic method. The hyperbolization idea is to transform the elliptic diffusive term into a hyperbolic one by introducing the diffusive flux as an additional variable. This new variable is thus the solution of an additional hyperbolic equation. The advantages of this method are: 1) the discretization used for the non-dissipative (inviscid) part of the equations can be directly applicable, 2) a speed-up factor of O ( 1 / h ) is obtained (h being the typical mesh spacing), 3) the diffusive terms are computed to the same order of accuracy as the main solution. We also consider the two-dimensional incompressible Navier-Stokes equation, with the incompressibility of the flow being ensured via the artificial pseudo-compressibility technique. The hyperbolic technique has been successfully applied to diffusion, advection-diffusion, Navier-Stokes, and magnetohydrodynamic equations [
The paper is organized as follows. Section 2 is devoted to the model and numerical method. In Section 3, the validation of the scheme and tests are presented for the single-sided cavity case. The results for the four-sided lid-driven cavity are presented in Section 4, with a particular emphasis on the bifurcation of the steady-state flow solution. Finally, conclusions are drawn in Section 5.
The time dependent Navier-Stokes (NS) equation is generally written as,
∂ v ∂ t + ( v ⋅ ∇ ) v + ∇ P − ν ∇ 2 v = 0 , (1)
for the velocity flow v , where P is the kinematic thermal pressure (i.e. thermal pressure divided by the fluid density), and ν is the kinematic viscosity coefficient.
Thus, the steady-state (i.e. ∂ ∂ t ≡ 0 , t being the time) Navier-Stokes equation follows,
( v ⋅ ∇ ) v + ∇ P − ν ∇ 2 v = 0. (2)
The incompressibility property must be also imposed via,
∇ ⋅ v = 0 . (3)
For an homogeneous viscosity, an equivalent conservative form of Equation (2) and Equation (3) is,
div ( v ⊗ v + P I ¯ ¯ ) − div ( ν g ¯ ¯ ) = 0, (4)
with g ¯ ¯ = grad v being the velocity gradient tensor, and where I ¯ ¯ is the identity rank-2 tensor. The last equation makes appear two flux tensors, the advective flux modified by the pressure effect F ¯ ¯ a = v ⊗ v + P I ¯ ¯ which can be also called the inviscid flux below, and the viscous stress tensor F ¯ ¯ v = − ν g ¯ ¯ . The full flux is thus F ¯ ¯ = F ¯ ¯ a + F ¯ ¯ v .
The previous incompressible Navier-Stokes equation can be solved by introducing a pseudo-time parameter τ in the following compressible equations,
∂ P ∂ τ + div ( c 2 v ) = 0, (5)
and,
∂ v ∂ τ + div ( v ⊗ v + P I ¯ ¯ ) − div ( ν g ¯ ¯ ) = 0, (6)
with c2 the square value of c, an artificial sound speed. The value of c is arbitrarily chosen in order ensure that the steady-state solution of Equation (5) and
Equation (6) (i.e. ∂ ∂ τ ≡ 0 ) is equivalent to the originel incompressible
Navier-Stokes equation. In practical, the value of c is chosen to be one in this work. This pseudo-incompressible form is said to be equivalent to the incompressible NS equations in the steady-state [
It is possible to introduce another differential equation for the stress tensor as,
T r ∂ g ¯ ¯ ∂ τ − grad v = − g ¯ ¯ , (7)
where T r is the relaxation parameter defined by T r = L r 2 / ν , with L r a length-scale parameter. The value of L r value is determined in order to achieve an optimal fast convergence [
In two dimensions, the system 5 - 7 can be expressed in the following compact form,
∂ U ∂ τ + ∂ F ∂ x + ∂ G ∂ y = S , (8)
with S being a source term containing the g ¯ ¯ tensor, and where F and G represent x-directed and y-directed full flux respectively.
In order to fix the ideas, we can write a simpler one-dimensional 1D (x dependent) version below, where the fluid velocity flow is given by u ( x ) . Then, the set of equations becomes,
∂ P ∂ τ + ∂ ( c 2 u ) ∂ x = 0, (9)
∂ u ∂ τ + ∂ ( u 2 + P − ν g ) ∂ x = 0, (10)
∂ g ∂ τ − 1 T r ∂ u ∂ x = − g T r . (11)
In this one dimensional version, U is a column vector containing three components, the pressure P, the velocity u, and g the gradient of u. The corresponding flux F has three components that are, c 2 u , u 2 + P − ν g , and − u / T r . Finally, the source term S has three components, that are zero for the two first ones and − g / T r for the third one. It is important for the following numerical implementation to determine the eigenvalues of the inviscid flux Jacobian (i.e. ignoring the viscous terms), that are λ ± = u ± u 2 + c 2 .
We consider the 1D version below, as it is simpler to write. For a one-dimensional grid of N nodes with a uniform spacing h, the solution can be evaluated at the nodes denoted by x j , j = 1 , 2 , 3 , ⋯ , N . Our finite-volume cell-centered discretization is thus,
∂ U j ∂ τ = − 1 h [ F j + 1 / 2 − F j − 1 / 2 ] + 1 h ∫ I j S d x , (12)
where U j denotes the solution value at the node j, F j + 1 / 2 is an interface flux to be defined (see below), and I j = [ x j − 1 / 2 , x j + 1 / 2 ] is a dual cell.
Steady state solutions of the previous system (i.e. with ∂ ∂ τ ≡ 0 ) can be
obtained in different ways. The simplest way consists in marching towards the steady-state using a pseudo-time iteration with a small increment Δ τ . This is similar to a standard scheme (e.g. Euler or Runge-Kutta method) used to integrate in time a general differential equation using a marching procedure with a time-step Δ t . Moreover, in our case, the method is a relaxation procedure towards a steady-state, with an efficiency determined by the choice of the T r parameter (or the associated L r parameter). Steady state solutions of the previous system can be thus obtained by using the simply pseudo-time explicit iteration,
U j n + 1 = U j n − Δ τ R e s j n , (13)
where n is the iteration counter, Δ τ is the pseudo-time step, and − R e s j n is the residual right-hand side of Equation (12). The latter residual term is required to vanish (up to a pre-defined given accuracy) when a steady state solution is obtained.
The numerical fluxes can be estimated using the standard upwind formula,
F j + 1 / 2 = 1 2 ( F L + F R ) − 1 2 c M ( U R − U L ) , (14)
where the subscripts L and R stand for the left- and right-hand sides of the cell interface situated at x j + 1 / 2 respectively. The first term is computed from an average value of the two fluxes F L = F ( U L ) and F R = F ( U R ) . Instead of using a full flux directly, the full flux can be separated into two parts (inviscid and viscous parts), for which the above numerical formula can be used separately (see paper [
In the explicit pseudo-time iteration, the value of the increment in the pseudo-time variable τ is limited by a criterion,
Δ τ = C F L × M i n [ h c M + ν / L r ] , (15)
where CFL is the Courant-Friedrichs-Lewy number less than or equal to one, and where we have simply added the maximum wave speed to the dissipative wave speed in the denominator.
The left and right values can be evaluated by a linear extrapolation from the cell centre, as for example for the u solution of the 1D discretization,
u L = u j + 1 2 h ∇ u j , u R = u j − 1 2 h ∇ u j , (16)
where ∇ u j is the gradient of u j . A simple and economic method is to use g j to evaluate the gradient term, as g is the spatial derivative of u in the steady state. This is a second order scheme that has been previously shown to converge rapidly towards the steady state. Moreover, some classical limited reconstructions options can be easily added to the scheme, that are the superbee, Koren, monotized-central, or minmod limiters.
The previous one dimensional model can be also easily discretized in two dimensions. The resulting set of hyperbolic equations can be obtained from the work done by Nishikawa (2011) (see [
As a test case, we first consider the one sided square cavity, where only the upper plate is moving with a velocity V x = V u = 1 (see
Note that a minmod limiter is chosen for this run. A grid resolution of 130 × 130 uniformly spaced cells is also employed, that is shown to lead to well converged results (see below). Moreover, we have checked that using a higher grid resolution of 180 × 180 does not modify the results. We have also found that
taking L r = 1 20 π , allows an optimal and fast convergence of our results towards
steady-state solutions. The stream lines solution plotted in
In order to check the validity and convergence of our results, we have plotted in
and R e = 100 . Note that an optimal value of L r = 1 20 π is chosen in order to
ensure a fast convergence of the results. Our results are compared with the results obtained by Ghia et al. (1982) that is the usual numerical solution taken as the reference in the litterature [
We focus now on the four-sided square cavity case, where the velocity value imposed at each plate is taken to be one. In
Now, we explore different Reynolds number values by increasing it over the previous value of R e = 100 . The results of the residual term evolution as a function of the iteration counter are plotted in
the critical value of the Reynolds number R c for the bifurcation. We precisely get R c = 129.4 with our procedure.
In this study, we use the hyperbolic method in order to study steady-state flow solutions of the four-sided lid-driven square cavity in two dimensions. A classical Riemann solver is used to solve the incompressible Navier-Stokes equation with the artificial compressibility method. A simple second-order finite-volume discretization on a rectangular grid using upwind advective and dissipative fluxes is employed. Our results show a remarkable efficiency to determine the critical Reynolds number R c corresponding to the bifurcation between a single stable solution at a low Reynolds number to a doubly stable one when the Reynolds number is increased. This is precisely obtained by the examination of the convergence behavior of the residual terms during the explicit pseudo-time iteration. We get the critical value R c = 129.4 , which is in the range of values [ 129 : 130.4 ] previously deduced in the literature.
The author declares no conflicts of interest regarding the publication of this paper.
Baty, H. (2022) Hyperbolic Method to Explore Multiplicity Flow Solutions in a Four-Sided Lid-Driven Cavity. American Journal of Computational Mathematics, 12, 314-323. https://doi.org/10.4236/ajcm.2022.123021