Dynamics of two-dimensional square lid-driven cavity at high Reynolds number

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Dynamics of two-dimensional square lid-driven cavity at high Reynolds number

By J.-Ch. Robinet, X. Gloerfelt and Ch. Corre

SINUMEF Laboratory, ENSAM CER de PARIS 151, Boulevard de l’Hˆopital,

75013 PARIS, FRANCE

email : [email protected]

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Generalities

Objective:

to precisely describe the space-time dynamics of a two-dimensional lid-driven cavity for various Reynolds numbers.

Two methods:

by solving the 2-D incompressible Navier-Stokes equations with high-order nu- merical scheme,

by solving a linear BiGlobal stability equations.

Explanatory sketch of square lid-driven cavity.

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Numerical algorithms (1)

Accurate projection method for nondimensional Navier-Stokes equations

(Brown et al., JCP 2001)

:

u^∗ − uⁿ

∆t + [(u.∇)u]ⁿ⁺¹² = −∇pⁿ⁺¹² + 1

Re

uⁿ⁺¹ +uⁿ 2

uⁿ⁺¹ is recovered from the projection of u^∗ :

∇.uⁿ⁺¹ = 0 ⇒ φⁿ⁺¹ = ∇.u^∗

∆t Poisson pb with four Neumann BC (singular problem) projection-updates :

uⁿ⁺¹ = u^∗ − ∆t∇φⁿ⁺¹ pⁿ⁺¹² = pⁿ⁻¹² + φⁿ⁺¹− ∆t

2Reφⁿ⁺¹ N(uⁿ⁺¹²) = [(u.∇)u]ⁿ⁺¹² =

4 j=1

βjN(u^n+1−j) third-order Adams-Bashforth scheme

→ second-order accuracy in time

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Numerical algorithms (2)

Space discretization:

slowly non-uniform Cartesian grid with a staggered arrangement

tridiagonal compact ﬁnite-diﬀerence scheme on a 7 point-stencil with sixth-order accuracy on arbitratry non-uniform Cartesian grids

sixth-order Lagrange interpolation calculated on the non-uniform grid

Semi-implicit time integration:

Cranck-Nicolson for viscous terms and Adams-Bashforth for convective terms

Non-iterative Poisson solver:

direct solution of the block-tridiagonal matrix from the Poisson equation with Thomas’ algorithm

matrix inversion based on LU-decomposition (7 GFlops on Nec SX-5) → designed for DNS of non-stationary ﬂows

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Regularization problems

A Poisson problem with 4 Neumann BC is singular :

Tikhonov regularization at the last substep of Thomas’ algorithm.

To obtain regularized solution to Ax = y, choose x to ﬁt data y in least-squares sense, but penalize solutions of large norm. Solve minimization problem

xα = min

x∈XAx − y²_Y +αx²_X = (A^∗A+ αI)⁻¹A^∗y

Substraction of corner singularities thanks to asymptotic expansions of the Stokes solutions at the two upper corners A & B.

Botella & Peyret (C&F 1999); Auteri et al.(JCP 2002)

singular solution for A :

uÂ_Re = uÂ₀ + ReuÂ_i pÂ_Re = pÂ₀/Re +pÂ_i uÂ₀ : Stokes problem uÂ_i : inertial component

U(x, t) = u(x, t) + u^A_Re(x) + u^B_Re(x)

P(x, t) = p(x, t) + p^A_Re(x) + p^B_Re(x) solve for u, p

ϕ^A_Re +ϕ^B_Re, Re = 100

A B

X

Y

0 0.2 0.4 0.6 0.8 1

→ no great inﬂuence of corner-singularity substractions because of the staggered arrangement

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BiGlobal Analysis: generalities (1)

Instantaneous physical quantities : Q(x, y, z, t) = Q(x, y) +εq(x, y)exp(iβz−ωt) +c.c., ε 1 the basic ﬂow Q(x, y) is supposed to be steady, two-dimensional (homogeneous in z direction)

and solution to 2-D incompressible Navier-Stokes equations.

q(x, y) are the eigenfunctions where β = 2π/λ_z is z-wavenumber, ω is the complex circular frequency (eigenvalues).

Im(ω) < 0 : Stable, Im(ω) > 0 : unstable. Re(ω) = 2πf : frequency.

Linearized Navier-Stokes equations become a partial diﬀerential system:











Dxu +Dyv + βw = 0, L − DxU

u − (DyU)v − Dxp = −ωu,

−(DxV )u + L − DyV

v − Dyp = −ωv,

Lw +βp = −ωw,

where

L = N − UDx − VDy and N = (1/Re) D_x² + D²_y −β²

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BiGlobal Analysis: generalities (2)

Boundary conditions:

For this diﬀerential system, the boundary conditions u = v = w = 0 are imposed on the perturbation velocity components at the walls.

Boundary conditions for the disturbance pressure do not exist physically ; compatibility conditions are used:

∂p

∂x = 1

Re∆_2du − U∂u

∂x − V ∂u

∂y, and ∂p

∂y = 1

Re∆_2dv − U∂v

∂x − V ∂v

∂y Numerical considerations:

Spectral collocation method is used (Chebyshev polynomials with Gauss-Lobatto points).

The diﬀerential system becomes an algebraic eigenproblem:

AX = ωX where ω are eigenvalues and X are eigenvectors.

The eigenproblem is solved by an QZ algorithm if A is not too large and Arnoldi algorithm if A is large.

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BiGlobal Analysis: Basic Flow (3)

Deﬁnition 1 (Basic Flow)

A basic flow is a solution of the equations of the movement with the greatest degree of symmetry compatible with the geometry of the problem.

In the lid-driven cavity case, the basic ﬂow is computed by 2-D Navier-Stokes equations.

If the solution is stationary then the basic ﬂow is this solution.

If the solution is not stationary then the ”basic ﬂow” is the mean solution.

We obtained basic flow solutions on square finite difference grids using high resolution inaccessible to the instability analysis and used a cubic spline interpolation scheme (NAG library) to interpol the basic flow solution on to the stability analysis grid.

The basic ﬂow solutions are converged in time to within a error err ≡ |(E_t₀+∆t − E_t₀)/E_t₀| <

10⁻¹⁰, where E is an integral measure (energy) of the ﬂow.

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Basic Flow: Reynolds Number Inﬂuence (1)

X

Y

0 0.2 0.4 0.6 0.8 1

X

Y

0 0.2 0.4 0.6 0.8 1

X

Y

0 0.2 0.4 0.6 0.8 1

X

Y

0 0.2 0.4 0.6 0.8 1

Contours of the steady streamfunction for various Reynolds numbers Re: from left to the right and from top to bottom: Re=100, 1000, 5000 and 10000.

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BiGlobal Results: critical Reynolds number

Basic ﬂow is stable when the Reynolds number < 8135. (stability grid used: 57 × 57)

At Re = 8135 the eigenvalue is equal to ω = 2.80637−1.6945×10⁻⁴i, (f = Re(ω)/2π = 0.4466) The bifurcation is super-critical: Amplitude ∼ √

Re − Re_c (DNS results).

Im(‘w)

Re(‘w)

-3 -2.5 -2 -1.5 -1 -0.5 0 0.5

-3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3

Re(‘w)

Im(‘w)

-2.82 -2.815 -2.81 -2.805 -2.8 -2.795 -2.79 -0.01

-0.005 0 0.005 0.01 0.015

Rey=8000 Rey=8100

Rey=8135

Rey=8500

(a) Discretized eigenvalue spectrum at Re = 8135.(b) Growth rate evolution versus Reynolds number.

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BiGlobal Results: Eigenfunctions

Eigenfunctions of most unstable eigenvalue at Re = 8135.

X

Y

0 0.5 1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

X

Y

0 0.5 1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

a) Real part of critical streamlines eigenfunction. (b) Imaginary part of critical streamlines eigenfunction.

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DNS Results: critical Reynolds number

In DNS point of view, the ﬂow becomes unsteady for 7950 < Re_c < 8000.

In agreement with the various results already published, the threshold is very sensitive to the various numerical parameters. Peng et al. (7704), Fortin et al. (8000), Sahin & Owens (8069), Bruneau & Saad (8050), Auteri & Parolini (8018), Tiesinga et al. (8375) and Cazemier (7972).

0 0.5 1 1.5 2

x 10⁶

−7.5

−7

−6.5

−6

−5.5

−5

iterations

residual for u

0 0.2 0.4 0.6 0.8 1

10⁻²⁰ 10⁻¹⁵ 10⁻¹⁰ 10⁻⁵ 10⁰

f

PSD per Hz

Sensor in (1/12,1/12) Left: Residue for u, red: Re = 7900, green: Re = 7950 and blue: Re = 8000.

Right: power spectral density in log scale. Re = 8000, Grid: (250 ×250), u (red) and v (blue).

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Results: Re ∈

8 × 10 ³ , 10 ⁴

For higher Reynolds number, other Hopf bifurcations appear:

BiGlobal Results:

Re_c 8135 8731 9054 9135 10129

0.446 0.445 0.617 0.617 0.617 Greater 0.617 0.441 0.438 0.710 Re(ω)/2π 0.534 0.534 0.431 Im(ω)

0.715 0.530 0.766 Lower 2-D DNS Results: ”linear” frequencies

Re_c 8000 8500 9000 10000 0.45 0.445 0.442 0.617

0.618 0.436 Re(ω)/2π 0.534 0.52

0.70

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BiGlobal Results: Re = 10 ⁴ (1)

At Re = 10⁴ four modes are linearly unstable.

Im(ω) 0.093 0.070 0.055 0.051 Re(ω)/2π 0.61 0.71 0.43 0.53

Re(ω)

Im(ω)

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

-2 -1.5 -1 -0.5 0 0.5

Discretized eigenvalue spectrum at Re = 10⁴.

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BiGlobal Results: Re = 10 ⁴ (2)

x

y

0 0.2 0.4 0.6 0.8 1

x

y

0 0.2 0.4 0.6 0.8 1

x

y

0 0.2 0.4 0.6 0.8 1

x

y

0 0.2 0.4 0.6 0.8 1

Real part of u from the most unstable to less unstable eigenmodes.

x

y

0 0.2 0.4 0.6 0.8 1

x

y

0 0.2 0.4 0.6 0.8 1

x

y

0 0.2 0.4 0.6 0.8 1

x

y

0 0.2 0.4 0.6 0.8 1

Imaginary part of u from the most unstable to less unstable eigenmodes.

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DNS Results: Re = 10 ⁴ (1)

At Re = 10⁴, the ﬂow is quasi-periodic with the main frequencies close to those obtained by stability analysis. The frequency f = 0.61 is the dominant mode.

However the amplitudes of some modes are diﬀerent. The second and third modes in DNS are f = 0.43 and f = 0.174 respectively . These diﬀerences can be interpreted by a transfer of energy of the fundamental frequencies towards the low frequency harmonics, characteristic of a strongly nonlinear regime.

300 302 304 306 308 310

−3

−2

−1 0 1 2 3x 10⁻³

t

u, v

0 0.2 0.4 0.6 0.8 1

10⁻²⁰ 10⁻¹⁵ 10⁻¹⁰ 10⁻⁵ 10⁰

f

PSD per Hz

Sensor in (1/12,1/12) Left: time evolution of u (red) and v (blue). Right: power spectral density in

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DNS Results: Re = 10 ⁴ (2)

The DNS ﬁnds the four frequencies predicted by the stability analysis.

An analysis of a higher nature (bispectra) permits to highlight the interactions between the main frequencies and to ﬁnd the value of the secondary frequencies.

Frequencies Main Frequencies Theoretical Values

0.09 f2 −f3 0.09

0.174 f2 −f1 0.18

0.263 2f1 −f2 0.262 0.347 2f2 −2f1 0.348

0.44 f1 0.436

0.52 f3 0.52

0.61 f2 0.61

0.70 f4 0.70

0.783 2f2 −f1 0.784

0.874 2f1 0.872

0.957 3f2 −2f1 0.958

1.047 f1 +f2 1.048

1.22 2f2 1.22

Bispectra of ﬂuctuating velocity u.

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Prospects (1)

Future works on 2-D cavity:

determination of the speciﬁes value of the critical Reynolds numbers, determination of the stability of the various periodic branches,

study of chaotic dynamics to high Reynolds number (Re ≥ 10000).

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Prospects (2)

Future works on 3-D cavity:

The dynamics of the three-dimensional cavity (periodic conditions in the spanwise direction) is qualitatively diﬀerent: existence of Taylor-G¨ortler instability for Re ≥ 780 (Theofilis et al.2004).

X

0 0.2 0.4 0.6 0.8 1

Y 0.2 0 0.6 0.4 1 0.8 Z

0 0.2

0.4

X

Y Z

ISOVORTICITES EN FLUCTUATIONS POUR LE MODE S1 (Valeurs: 0.22, 0.66, 2.06)

X

0 0.2 0.4 0.6 0.8 1

Y 0.2 0 0.6 0.4 1 0.8 Z

0 0.2

0.4

X

Y Z

ISOVORTICITES EN FLUCTUATIONS POUR LE MODE T1 (Valeurs: 0.39, 0.88, 3.14)

X

0 0.2 0.4 0.6 0.8 1

Y 0.2 0 0.6 0.4 1 0.8 Z

0 0.2

0.4

0.6 0.8

X

Y Z

X

0 0.2 0.4 0.6 0.8 1

Y 0.2 0 0.6 0.4 1 0.8 Z

0 0.2

0.4

X

Y Z

Spatial distribution of the magnitude of disturbance vorticity of the most unstable eigenmodes in the square lid-driven cavity. From left to right: mode S1 (Re = 1000, β = 15), mode T1 (Re = 1000,

β = 15), mode T2 (Re = 1000, β = 7.5) and mode T3 (Re = 1000, β = 15) .

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Nonlinear wave interaction

DSP provide no phase information

⇒ High-Order Spectra (HOS)

Third-order spectrum or Bispectrum :

B(f

i

, f

j

) = lim

T→∞

1 T E [X (f

i

)X (f

j

)X

^∗

(f

i

+ f

j

)]

A peak indicates a nonlinear interaction between fi and fj

Previous applications to cavity ﬂow :

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Basic Flow: Validations (2)

X U

V Y

0 0.2 0.4 0.6 0.8 1

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

0 0.2 0.4 0.6 0.8 1

X U

V Y

0 0.2 0.4 0.6 0.8 1

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

0 0.2 0.4 0.6 0.8 1

(a) Reynolds number 1000. (b) Reynolds number 10000.

Comparison of velocity profiles obtained on the median planes: x = 0.5 and y = 0.5, in red geometrical grid 301× 301, in green 101 × 101, black points : Ghia, points red: Erturk.

Dynamics of two-dimensional square lid-driven cavity at high Reynolds number