Approximation of pressure perturbations by FEM

The mathematical problem of linear hydrodynamic stability is [1] ∂u

and its solution is the perturbation (u ′ , p ′ ), for t > 0. Here all variables are non dimensional, (U, P ) is the basic motion in the domain Ω ⊆ R n , n = 2 or 3, Re = U ∞ L/ν is the Reynolds number, L is a length scale, U ∞ is a velocity scale, ν is the coefficient of kinematic viscosity.

Applying the divergence operator to (1a) and using the divergence-free condition on u ′ , we obtain

and projecting (1a) on n * , we have

where n * is the unit outward normal to ∂Ω and

Another form of the right hand -side of the Poisson equation for p ′ can be obtained by using the divergence-free condition on U.

Thus the mathematical problem of the linear hydrodynamic stability becomes (1a), ( 2), (1c), ( 3), (1d). Of course, it is assumed that equation (1a) can be continued on the ∂Ω.

Let Ω = {x = (x, y, z) ∈ R 3 | -∞ < x, z < ∞, 0 < y < a} (a ≥ 1) and assume that the basic flow is of the form U(x, y, z) = (U (y), 0, 0). Let us choose Tollmien-Schlichting waves -like perturbations u ′ (t, x, y, z) = u ′ 0 (y)exp[iα(x-ct)], u ′ 0 (y) = (u(y), v(y), 0), u ′ (t, x, y, z) = (u ′ (t, x, y), v ′ (t, x, y), 0), p ′ (t, x, y, z) = p(y)exp[iα(x -ct)], where i = √ -1, α is the streamwise wave number, c = c r + ic i , c r is the wave speed and c i is the amplification rate.

In this case, model (1) leads to the classical two -point eigenvalue problem for Orr -Sommerfeld equation in (c, ϕ), where ϕ is the nonexponential factor of the stream function, while (1a), (2), (1c), (3), (1d) becomes the two -point eigenvalue problem in (c, u, v, p)

v(y) = 0 for y = 0, y = a, (4e)

where the prime stands for the differentiation with respect to y, c is an eigenvalue and (u, v, p) is an eigenvector.

Let L 2 (0, a) be the Hilbert space of all measurable complex-valued functions u, defined on (0, a), for which

uvdy. Denote Du the generalized derivative and consider the spaces

Multiply (4a), ( 4b) and (4c) by arbitrary test functions f 1 , f 2 and g respectively, integrate over (0, a), apply partial integration if necessary and take into account the two point conditions (4d) -(4f) to obtain the following weak formulation of problem ( 4

3 Approximation of problem ( 5) -( 7) by FEM

In order to perform this approximation, let us divide the interval [0, a] in N + 1 subintervals

The approximate basic shear flow U h (x, y, z) = (U h (y), 0, 0) is defined by U h (y) = U (y), y ∈ [0, a]. The approximate amplitudes u h (y), v h (y) and p h (y) correspond to the exact ones, u(y), v(y) and p(y), respectively.

Correspondingly, variational problem ( 5) -( 7) is approximated by the following problem in

In order to obtain u h , v h , we use a basis of real functions of V h . Let J = J K = {1, 2, 3} be the local numeration for the nodes of K, where 1, 3 correspond to y j , y j+1 respectively and 2 corresponds to a node between y j and y j+1 . Let {φ n , n ∈ J} be the local quadratic basis of functions on K corresponding to the local nodes. Let J * = {1, . . . , 2N + 1} be the global numeration for the nodes of [0, a], where the nodes corresponding to y 0 and y N +1 are not taken into account, and let L 1 be a matrix whose elements are the elements of J * . Its rows are indexed by the elements K ∈ T h and its columns, by the local numeration n ∈ J. We take the value 0 at the locations of L 1 which we do not consider in the computations, i.e. the locations where K = K 0 , n = 1 and K = K N , n = 3. Write n * = L 1 (K, n) or, simply, n * for the element n * of L 1 which depends on K and n. Let {(Φ n * , 0), (0, Φ n * ); n * ∈ J * } be a basis of functions of V 2 h . If n * corresponds to y j , then Φ n * is a real quadratic function on K j-1 and K j and its value is zero on [0, a](K j-1 ∪ K j ). If n * lies between y j and y j+1 , then Φ n * is a real quadratic function on K j and its value is zero on [0, a]\K j . We have Φ n * (y) = φ n (y), where y ∈ K, n * = L 1 (K, n). Let u n * , v n * be the values of u h , v h at the nodes n * , n * ∈ J * . Retaining our convention about n * = L 1 (K, n), we do not write in the sequel the conditions n = 1 for K = K 0 and n = 3 for K = K N . We have

and a similar expression for v h (y).

In order to obtain p h , we use a basis of real functions of M h . Let I = I K = {1, 2} be the local numeration for the nodes of K, where 1, 2 correspond to y j , y j+1 respectively. Let {ψ m , m ∈ I} be the local affine basis of functions on K corresponding to the local nodes. Let I * = {0, 1, . . . , N, N + 1} be the global numeration for the nodes of [0, a] and let L 2 be a matrix whose elements are the elements of I * . Its rows are indexed by the elements K ∈ T

for all k ∈ J, ℓ ∈ I, for all

represented by relations (11) -(13) can be written in the following matrix form

where K h , S h ∈ C 2(2N +1)×2(2N +1) , L h ∈ C 2(2N +1)×(N +2) , G h ∈ C (N +2)×(N +2) , H h ∈ C (N +2)×2(2N +1) . Matrix G h is posi