A class of high-resolution algorithms for incompressible flows Long Lee Department of Mathematics University of Wyoming Laramie, WY 82071, USA Abstract We present a class of a high-resolution Godunov-type algorithms for solving flow problems governed by the incompressible Navier-Stokes equations. The algorithms use high-resolution finite volume methods developed in SIAM J. Numer. Anal., 33, (1996) 627-665 for the advective terms and finite difference methods for the diffusion and the Poisson pressure equation. The high-resolution algorithm advects the cell-centered velocities using the divergence-free cell edge velocities. The resulting cell-centered velocity is then updated by the solution of the Poisson equation. The algorithms are proven to be robust for constant- density flows at high Reynolds numbers via an example of lid-driven cavity flow. With a slight modification for the projection operator in the constant-density solvers, the algorithms also solve incompressible flows with finite-amplitude density variation. The strength of such algorithms is illustrated through problems like Rayleigh-Taylor instability and the Boussinesq equations for Rayleigh-B´enard convection. Numerical studies of the convergence and order of accuracy for the velocity field are provided. While simulations for two-dimensional regular-geometry problems are presented in this study, in principle, extension of the algorithms to three dimensions with complex geometry is feasible. keywords: high-resolution, finite volume methods, incompressible Navier-Stokes equations, finite-amplitude density variation, lid- driven cavity flow, Rayleigh-Taylor instability, Rayleigh-B´enard convection 1 Introduction Chorin, in a series of papers [7–9] introduced a practical fractional-step method for solving viscous incompress- ible Navier-Stokes equations. Similar ideas were independently introduced by Temam [29]. This fractional-step method, or projection method, computes an intermediate velocity without regard to the divergence constraint and then projects this velocity onto the divergence-free subspace. Many other different forms of projection methods were developed after those of Chorin and Temam, making it impossible to report an exhaustive ref- erence list. We review the following two projection methods that are closely related to the proposed high- resolution algorithms. Consider the dimensionless incompressible Navier-Stokes equations with an external force 1 u + (u · r)u + rp = r2u + F ; t Re r · u = 0; (1.1) u = b on @Ω: The first projection method we consider is a finite difference method introduced by Kim and Moin [15]. This method is similar to the first-order scheme of Chorin. The method first computes the intermediate velocity 1 without the gradient pressure term. Then it imposes the incompressibility by solving a Poisson equation. The method can be written in a semi-discrete form: u∗ − un 1 + (un+1=2 · r)un+1=2 = (r2u∗ + r2un) + F ; (1.2) ∆t 2Re u∗ = b + ∆trφn on @Ω; (1.3) un+1 − u∗ = −∇φn+1; (1.4) ∆t r · un+1 = 0; (1.5) n · un+1 = n · b on @Ω; (1.6) where n is the unit normal. The nonlinear convection term (un+1=2 · r)un+1=2 can be approximated, for 3 n n 1 n−1 n−1 n+1 example, by an explicit Adams-Bashforth formula 2 (u · r)u − 2 (u · r)u . The pressure p can be obtained from φn+1 through the relationship [2] ∆t pn+1 = φn+1 − r2φn+1: (1.7) 2Re To compute φn+1, we apply the divergence operator ∇· to (1.4) and use (1.5). We then have the Poisson equation r · u∗ r2φn+1 = : (1.8) ∆t We refer to this projection method as Kim and Moin’s scheme. The second method we consider is the pressure-correction scheme first introduced by Van Kan [31], and later Bell, Colella, and Glaz (BCG) modified the scheme by using Godunov’s methodology to compute the advection term [1]. We outline this projection method in the semi-discrete form as follows: u∗ − un 1 + (un+1=2 · r)un+1=2 + rpn−1=2 = (r2u∗ + r2un) + F ; (1.9) ∆t 2Re u∗ = b on @Ω; (1.10) un+1 − u∗ rpn+1=2 − rpn−1=2 = − = −∇φn+1; (1.11) ∆t γ r · un+1 = 0; (1.12) n · un+1 = n · b on @Ω: (1.13) From the first equality of (1.11), we have (∆t2) @rp un+1 ≈ u∗ − : (1.14) γ @t Equation (1.14) implies u∗ = b + O(∆t2) on @Ω. Discussion on the choice of γ can be found in [13, 27]. We refer to this project method as the BCG scheme. The projection method introduced for the high-resolution algorithms can be put into either the framework of Kim and Moin’s scheme or the BCG scheme. 2 2 Conservative algorithms for advection The class of high-resolution algorithms for incompressible flows proposed in this study is motivated by the high- resolution wave-propagation algorithms for advection in incompressible flows, developed by LeVeque [18]. A brief introduction for the wave-progation algorithm is described as follows. We consider the scalar advection equation in a specified velocity field u(x; t) in one, two, or three space dimensions qt + r · (uq) = 0; (2.1) where q(x; t) is a conservative quantity which can be the scalar concentration or the density function. If the wflowhere isq(x incompressible,, t) is a conservrativ · euq=ua 0n,ti thety w conservativehich can be t formhe sc (2.1)alar c canonce bentr writtenation or inth ae advectivedensity fun formction. If the flow is incompressible, u = 0, the conservative form (1.3) can be written in a advective form ∇ · qt + u · rq = 0: (2.2) q + u q = 0. (1.4) t · ∇ vi,j+1/2 Qi,j PSfrag replacementsui−1/2,j ui+1/2,j vi,j−1/2 Figure 1.1: Cell-variable locations for the conservative algorithm. Figure 2.1: Cell-variable locations for the conservative algorithm. Although (1.3) and (1.4) are mathematically equivalent for incompressibe flow, numerical algorithms based on theAlthoughtwo may (2.1)behave anddiff (2.2)erently are. In mathematicallytwo dimensions equivalent, a staggered forgr incompressibleid as shown in F flow,igure 1 numerical.1 is used i algorithmsn [?] sobasedthat th one n theume tworical mayalgo behaverithms m differently.odeled eithe Inr f tworom dimensions,the conservat theive staggeredform or fro gridm th ase a showndvective info Figurerm are 2.1 is the same. The notations in Figure 1.1 are as follows. Let C be the (i, j) grid cell [x , x ] used in [18] so that the numerical algorithms modeled eitherij from the conservative formi−1/ or2 fromi+1/2 the× advec- [yj−1/2, yj+1/2 ]. The “edge velocities” ui±1/2,j and vi,j±1/2 are the velocities at the midpoints of the inter- tive form are the same. The notations in Figure 2.1 are described as follows: Let Cij be the (i; j) grid cell faces (xi±1/2, yj) and (xi, yj±1/2), giving rise to the waves being propagated. They should satisfy the discrete di[vxeir−g1e=n2c;e x-fir+1ee=2re]la×tion[ysjh−ip1=2; yj+1=2 ]. The “edge velocities” ui±1=2;j and vi;j±1=2 are the velocities at the mid- points of the interfaces (xi±1=2; yj) and (xi; yj±1=2), giving rise to the waves being propagated. They should ui+1/2,j ui−1/2,j vi,j+1/2 vi,j−1/2 satisfy the discrete divergence-free relationship− + − = 0. (1.5) x y $ $ ui+1=2;j − ui−1=2;j vi;j+1=2 − vi;j−1=2 Qi,j represents an approximation to the cell average, + = 0: (2.3) ∆x ∆y 1 Q q(x, y, t) dx dy. n (1.6) Qi;j represents an approximation toi, thej ≈ cellx averagey at the current time level t , $ $ !Ci,j Z Using the relationship (1.5) and the upwind wave-pr1opagation method, LeVeque [?] developed the high-resolution Qi;j ≈ q(x; y; t) dx dy: (2.4) conservative algorithms for (1.4) in incompress∆iblxe∆flyow.Ci;jIn these algorithms, a flux-limiter is applied to the second-order Lax-Wendroff method to deal with steep gradients or even discontinuities in q(x, y, t) without inIntro twoducin dimensiong spurious space,numer weical consideroscillatio thens scalaror exc color-equationessive numerica (2.2)l diff inusi theon. componentThis high-r form,esolution property gives our fractional step method the edge in handling flows where the conserved quantities have a sharp dis- continuity between fluids, such as the denqstit+y iun(x;Ra yy)leqixgh+-Tva(yx;lor y)inqysta=bi 0li:ty . In two dimensional space, the (2.5) high-resolution algorithms can use either dimensional split or unsplit methods. The unsplit method improves ? the first-order corner transport upwind (CTU) [ ] by adding3 correction waves and transverse propagation of the correction waves. 2 The unsplit finite volume wave propagation algorithm for solving the above equation takes the form ∆t Qn+1 =Q − A+∆Q + A−∆Q i;j i;j ∆x i−1=2;j i+1=2;j ∆t + − − B ∆Q + B ∆Q (2.6) ∆y i;j−1=2 i;j+1=2 ∆t ∆t − F~ − F~ − G~ − G~ : ∆x i+1=2;j i−1=2;j ∆y i;j+1=2 i;j−1=2 + The term A ∆Qi−1=2;j represents the first-order Godunov update to the cell value Qi;j resulting from the Riemann problem at the edge (i − 1=2; j). The other three similar terms are the Godunov updates resulting from the Riemann problems at the other three edges.