Modeling of optimal perturbations in flat plate boundary ... - Ensam

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F. Alizard, J.-C. RobinetFollowing the development of Barkley et al. [8], we can associate the action of the operator A (t) to the temporalintegration of the linearized Navier–Stokes equations (4).The action of the operator A ( t f) + can be evaluated by introducing a second scalar product:〈q, q+ 〉 =∫ t ft 0∫Dq t Bq + dDdτ (14)The adjoint system of (4) according to (14) is thus written as follows:⎧⎪⎨ B ∂q+ = A + q +∂t⎪⎩ q + ( (15))x, t = t f = q+t fwith⎛⎞D + C − ∂U/∂x −∂V /∂x −∂/∂xA + = ⎜ −∂U/∂y D + C − ∂V /∂y −∂/∂y ⎟⎝⎠ , (16)∂/∂x ∂/∂y 0where the required homogeneous boundary conditions and relationship between initial data for direct andadjoint systems:⎧⎨ u = u + = v = v + = 0inδD⎩and [( u, u +)] t ft 0= 0inDThe solution to the adjoint system (15) leads to a similar evolution operator as the one regarding the directsystem referenced as:u + (x, t 0 ) = A + ( t f)u+ ( x, t f)(18)A + ( ) ( )t f is the adjoint of A t f under the inner product (7), since:( ( )A t f u (t0 ) , u + ( )) ( ( )t f = u t f , u+ ( )) (t f = u (t0 ) , u + (t 0 ) ) =(17)(u (t 0 ) , A ( t f) + u+ ( t f) ) (19)Thus, an expression for (8) may be expressed in a vorticity–streamfunction formulation by taking the largesteigenvalue of the generalized eigenproblem:L + A + ( t f)A(t f)Lg0 = λL + Lg 0 (20)The optimal perturbation u opt which maximizes G ( t f)is thus given by the corresponding eigenvector Lg0 .An efficient way to compute u opt is to use a power iterations algorithm:Lg n+1 = ρ n L ( L + L ) −1 L + A + ( t f)A(t f)Lgn (21)where ρ n is an arbitrary scaling parameter. Finally, to compute the optimal growth we must act with the operatorL ( L + L ) −1 L + A + ( t f)A(t f)L. This is accomplished by fixing a relationship between the forward anddirect solution. As detailed in [8], we impose the initial condition for the adjoint problem equal to the solutionunderlying the forward problem at time t f .Thereby, we derived an expression in vorticity–streamfunction formulation of the computation of the optimaldisturbances. We are then able to compute the quantity (8) by using a linearized version of the DNS codedescribed above. In practice, the optimal perturbation is computed by performing numerical time marching ofthe direct and adjoint system in vorticity–streamfunction formulation. In all numerical experiments reportedin this manuscript, the power iterations converged quickly (less than 10 iterations).
Modeling of optimal perturbations3.3 Global modes expansion analysis3.3.1 Theoretical aspectsA solution of (4) may be decomposed into a basis formed by global modes:∞∑q (x, t) = K k (t) ˆq k (x) with K k (t) = ˜K k e −i ktk=1(22)with ˆq k (x) e −i kt the global modes. The complex circular frequency k drives the temporal evolution of themode k. In addition, when I( k ) is negative the global mode is damped in time otherwise the global modeis amplified. The spatial structure of each global mode is deduced from ˆq k (x). These latters are computed byinserting (22)into(4) which leads to the generalized eigenvalue problem:Aˆq =−iBˆq (23)Introducing a third scalar product as:where • is the complex conjugate.The quantity (8) is written as:with[̂q1 ,̂q 2]=∫DG ( t f)= maxq 0E(q(t f ))E(q 0 )E(q) = 1 ] [̂q,̂q2̂q t 1 B̂q 2dD (24)(25)(26)It is thus more convenient to compute (25) with the Euclidean norm:G ( t f)=∥ ∥Fexp(t f D ) F −1∥ ∥ 2 2(27)where F represents the Cholesky decomposition of the inner product matrix of components [ˆq j , ˆq i]and D,the diagonal matrix of eigenvalues defined by D k,l =−iδ k,l k . 1 As a consequence, the computation of themaximum singular value of ∥ ∥Fexp ( t f D ) F −1∥ ∥2for each time t = t f allows to determine G (t) [26].Since the global modes basis is non-orthogonal, the biorthogonality condition may be used to determinethe components of the initial condition of the perturbation into this global modes expansion by:with 〈〈, 〉〉 a standard Hermitian scalar product.3.3.2 Numerical method˜K k =〈〈 ˆ q + k , q 0〉〉/〈〈ˆq + k , ˆq k〉〉 (28)The poor convergence of the spectrum requires a highly accurate numerical scheme in the two spatial directions[20]. Hence, the problem (23) is discretized with a Chebyshev/Chebyshev collocation spectral method.A spectral interpolation routine allows a transfer of the base flow from the DNS mesh to the stability mesh.Dirichlet conditions at the lower and upper boundaries and a specific Neumann condition based on a Gaster’stransformation both at the outflow and at the inflow [1,4,15] are employed to close the system (23). Moredetails on the inflow condition is presented in Appendix A. 2 Finally, a shift and invert Arnoldi algorithm fromthe ARPACK library provides the most important part of the spectrum [18].1 δ k,l is the Kroenecker symbol.2 The treatment of the pressure spurious modes is illustrated in [4,24]and[25].
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