26 Global 2D stability analysis of the staggered cavity
3.2. The Problem 27
The flow inside the cavity is governed by the unsteady 2D N-S equations which in non-dimensional form, can be written as
∂u
∂x +∂v
∂y = 0, (3.1a)
∂u
∂t +u∂u
∂x +v∂u
∂y =−∂p
∂x + 1
Re∇2u, (3.1b)
∂v
∂t +u∂v
∂x +v∂v
∂y =−∂p
∂y + 1
Re∇2v, (3.1c)
Hereuandv are the velocities alongx- and y-directions and pis the pressure;
Reis the Reynolds number defined asUL/ν, whereνis the kinematic viscosity of the fluid; U is the velocity of the lid and L= 1.4 is the width of the cavity which are used to non-dimensionalize the velocity and length-scale variables, respectively. For the present configuration, we have taken the non-dimensional velocity U as 1.
Introducing streamfunction ψ and vorticity ω, the above equations can be written as
ωt− 1
Re ωxx+ωyy
+ (uωx+vωy) = 0, (3.2) ψxx+ψyy =−ω(x, y). (3.3) with
u=ψy, and v =−ψx, (3.4)
and
ω =vx−uy. (3.5)
The boundary conditions for the antiparallel motion are given by ψ= 0 and ψy =−1 for y= 0, 0≤x≤1 ψ= 0 and ψx = 0 for x= 1, 0≤y≤0.4 ψ= 0 and ψy = 0 for y= 0.4, 1≤x≤1.4 ψ= 0 and ψx = 0 for x= 1.4, 0.4≤y≤1.4
(3.6)
28 Global 2D stability analysis of the staggered cavity
ψ = 0 and ψy = 1 for y= 1.4, 0.4≤x≤1.4 ψ = 0 and ψx = 0 for x= 0.4, 1≤y≤1.4 ψ = 0 and ψy = 0 for y= 1, 0≤x≤0.4 ψ = 0 and ψx = 0 for x= 0, 0≤y≤1
(3.7)
The boundary conditions for the parallel motion are same as (3.7), except that the first condition must be replaced by
ψ = 0 and ψy = 1 for y= 0, 0≤x≤1
These boundary conditions clearly indicate that flow singularities exist at the corners with coordinates (0,0), (1.0,0), (0.4,1.4) and (1.4,1.4).
3.2.1 The Numerical Scheme
In order to discretize the governing equations (3.2)-(3.3), we use the HOC scheme developed by Kalita et al. [56] for the unsteady convection-diffusion equation for variable convection coefficients. The scheme is briefly discussed below.
The unsteady two-dimensional convection-diffusion equation for a transport variable φ in some domain can be written as
a∂φ
∂t − ∇2φ+c(x, y, t)∂φ
∂x +d(x, y, t)∂φ
∂y =g(x, y, t) (3.8) whereais a constant,cand dare the convection coefficients, andg is a forcing function. The (9,9) HOC scheme by Kalita et al. [56] for this equation with accuracy O((△t)2, h4, k4) is given by:
3.2. The Problem 29
a
1 + h2
12(δx2−cijδx) + k2
12(δy2−dijδy)
δt+φnij − 1
2
αijδ2x+βijδ2y −Cijδx−Dijδy+ h2+k2
12 (δ2xδy2−cijδxδy2−dijδx2δy −γijδxδy)
(φnij +φn+1ij ) = 1
2(Gnij +Gn+1ij )
(3.9) where δt+ denotes the forward difference operator for time with uniform step length△t, and δx, δy are the first order andδx2, δy2 are the second order central difference operators in the space directions xand y with uniform step lengths h and k respectively.
The coefficients αij, βij, γij, Cij, Dij and Gij are given by:
αij = h2
12(c2ij −2δxcij) (3.10) βij = h2
12(d2ij −2δydij) (3.11) γij = 2
h2 +k2(h2δxdij +k2δycij −cijdij) (3.12) Cij = [1 + h2
12(δx2−cijδx) + k2
12(δ2y−dijδy)]cij (3.13) Dij = [1 + h2
12(δ2x−cijδx) +k2
12(δy2−dijδy)]dij (3.14) Gij = [1 + h2
12(δ2x−cijδx) + k2
12(δy2−dijδy)]gij (3.15) Here, it is assumed that the convection coefficients c, d and the forcing func- tion g are sufficiently smooth.
In order to solve the ψ-ω formulation of N-S equations using (3.9), we em- ploy an outer-inner iteration procedure and for reaching steady state, a time
30 Global 2D stability analysis of the staggered cavity
marching strategy is adopted. In a typical outer temporal cycle, we discretize equation (3.2) using (3.9) with a =Re, c=Reu, d=Rev and g = 0. Onceω is obtained, we compute ψ by discretizing (3.3) with the steady state version of (3.9). Thus on setting φ = ψ, c=d= 0 =δt+φij and g = −ω, the fourth order compact approximation of the Poisson equation (3.3) becomes
δ2x+δy2+ 1
12(h2 +k2)δ2xδy2
ψij =
1 + h2
12δx2+ k2 12δ2y
ωij. (3.16)
3.2.2 Approximation of the vorticity boundary condi- tions
The value of the streamfunctionψ is taken to be zero along all the boundaries while the vorticity boundary condition is derived using a fourth order compact scheme for the Neumann boundary condition. For example, on the leftmost wall (x = 0, 0 ≤ y ≤ 1), the approximation for ω can be found from the relationv =−[(∂ψ)/(∂x)] and equations (3.2), (3.4) and (3.5) as
−δx+ψ0j − h
2 +h2
6 δx+− h3
24(Rev0jδy−δy2)
ω0j =v0j − h3
24(δ+xδy2v0j −δ+t ω0j) (3.17) where the suffixes 0 and j stand for the leftmost wall and the vertical index respectively. Using equation (3.7) andv =−[(∂ψ)/(∂x)] we obtain an explicit expression forω on the leftmost wall as
ω0jn+1 = 24△t h3
−ψ1jn h −h
2ω0jn − h
6(ω1jn −ω0jn)− h
24(ω0j+1n −2ωn0j +ω0j−1n )
+ω0jn (3.18) Vorticities on the other walls can be found in a similar way and for the corners, we use a third order approximation as outlined in [105].
After discretization, the vorticity equation (3.2) and the streamfunction equa- tion (3.3) reduce to the single matrix form of the type
Aφ=B (3.19)
3.2. The Problem 31
where the coefficient matrix A is an asymmetric sparse matrix with each row containing at most nine non-zero entries, φ is the unknown vector (ψ or ω) and B is the vector of known (source) terms. For a grid of size m×n, Ais of size mn×mn, and φand B are mn-component vectors.
The next step now is to solve equation (3.19); as the coefficient matrixAis not generally diagonally dominant, conventional solvers such as Gauss-Seidel can- not be used. For both the vorticity and streamfunction equation, the resulting matrix equations are solved using BiCGStab [63, 101], which constitutes the inner iterations. Once (3.3) is solved, u and v in (3.4) are calculated using a fourth order compact formula (see ref. [56]). This constitutes one outer itera- tion cycle. For the inner iterations, the computations were stopped when the norm of the residual vector r =B−Aφ(φbeing either ψ orω) arising out of equation (3.19) fell below 10−6. Steady state was assumed to reach when the maximum ω-error between two successive outer temporal iteration steps was smaller than the tolerance limit 0.5×10−6.
3.2.3 The Algorithm
The algorithm for computing the flow inside the cavity by solving the ψ-ω formulation of the N-S equations can be summarized as follows:
1. Initialize ψ(n), ω(n), u(n) and v(n) for n = 0 with zero everywhere except at the boundaries where values from equations (3.7) and(3.17) are to be used.
2. Solve the vorticity transport equation of (3.2) by using the HOC formu- lation (3.9).
3. Solve the stream-function (ψ) equation of (3.3), using (3.16).
In both 2 and 3, solve the algebraic system of equations of the form (3.19) till the l2 norm of the residual, that is ||B−Aφ||2 < 10−6 (φ being either ψ orω).
4. Compute u(n+1) and v(n+1) from (3.4) using the fourth order compact approximation of Kalita et al. [56].
32 Global 2D stability analysis of the staggered cavity
5. If max|ω(n+1)−ω(n)|< 0.5×10−6 steady state is reached, stop compu- tation; else repeat steps 2 to 4 until convergence.
In the above (n+ 1) and (n) stand for the (n + 1)th and (n)th time levels, respectively.
Flow simulations are carried out using uniform grids of sizes 71×71, 85×85, 99×99, 106×106, 141×141 and 211×211 with a time step△t= 0.01 in the range of Reynolds numbers 50 ≤ Re ≤ 12,000 for both the antiparallel and parallel motion of the lid. In the cases where steady state is achievable, a time marching strategy is adopted till a steady state is reached. In all the computa- tions for this problem except in section 3.5, the data from the previous lower Reynolds numbers were used as the initial data except for Re = 50, where zero initial data was used. Unless otherwise mentioned, all the qualitative and quantitative results presented here are from computations on a 211×211 grid.