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� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � Zuccher, S., Tumin, A., Reshotko, E., Optimal Disturbances in Compressible Boundary Layers – Complete Energy Norm Analysis , Paper AIAA-2005-5314 – p. 5 x Optimal perturbation Boundary layer edge y Optimal perturbations Flat plate z most steady ini- which maximizes the energy given the of the a condition, energy perturbation? for disrupting, Question . is growth What initial tial
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� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � Zuccher, S., Tumin, A., Reshotko, E., Optimal Disturbances in Compressible Boundary Layers – Complete Energy Norm Analysis , Paper AIAA-2005-5314 – p. 5 x Optimal perturbation Boundary layer edge In this sense the perturbations are optimal. y Optimal perturbations Flat plate z most steady ini- which maximizes the energy given the of the a condition, energy perturbation? for disrupting, Question . is growth What initial tial
Goals/Tools Goals Zuccher, S., Tumin, A., Reshotko, E., Optimal Disturbances in Compressible Boundary Layers – Complete Energy Norm Analysis , Paper AIAA-2005-5314 – p. 6
Goals/Tools Goals Efficient and robust numerical determination of optimal perturbations in compressible flows. Zuccher, S., Tumin, A., Reshotko, E., Optimal Disturbances in Compressible Boundary Layers – Complete Energy Norm Analysis , Paper AIAA-2005-5314 – p. 6
Goals/Tools Goals Efficient and robust numerical determination of optimal perturbations in compressible flows. Formulation of the optimization problem in the discrete framework. Zuccher, S., Tumin, A., Reshotko, E., Optimal Disturbances in Compressible Boundary Layers – Complete Energy Norm Analysis , Paper AIAA-2005-5314 – p. 6
Goals/Tools Goals Efficient and robust numerical determination of optimal perturbations in compressible flows. Formulation of the optimization problem in the discrete framework. Coupling conditions automatically recovered from the constrained optimization. Zuccher, S., Tumin, A., Reshotko, E., Optimal Disturbances in Compressible Boundary Layers – Complete Energy Norm Analysis , Paper AIAA-2005-5314 – p. 6
Goals/Tools Goals Efficient and robust numerical determination of optimal perturbations in compressible flows. Formulation of the optimization problem in the discrete framework. Coupling conditions automatically recovered from the constrained optimization. Effect of energy norm choice at the outlet. Zuccher, S., Tumin, A., Reshotko, E., Optimal Disturbances in Compressible Boundary Layers – Complete Energy Norm Analysis , Paper AIAA-2005-5314 – p. 6
Goals/Tools Goals Efficient and robust numerical determination of optimal perturbations in compressible flows. Formulation of the optimization problem in the discrete framework. Coupling conditions automatically recovered from the constrained optimization. Effect of energy norm choice at the outlet. Tools Zuccher, S., Tumin, A., Reshotko, E., Optimal Disturbances in Compressible Boundary Layers – Complete Energy Norm Analysis , Paper AIAA-2005-5314 – p. 6
Goals/Tools Goals Efficient and robust numerical determination of optimal perturbations in compressible flows. Formulation of the optimization problem in the discrete framework. Coupling conditions automatically recovered from the constrained optimization. Effect of energy norm choice at the outlet. Tools Lagrange Multipliers technique. Zuccher, S., Tumin, A., Reshotko, E., Optimal Disturbances in Compressible Boundary Layers – Complete Energy Norm Analysis , Paper AIAA-2005-5314 – p. 6
Goals/Tools Goals Efficient and robust numerical determination of optimal perturbations in compressible flows. Formulation of the optimization problem in the discrete framework. Coupling conditions automatically recovered from the constrained optimization. Effect of energy norm choice at the outlet. Tools Lagrange Multipliers technique. Iterative algorithm for the determination of optimal initial condition. Zuccher, S., Tumin, A., Reshotko, E., Optimal Disturbances in Compressible Boundary Layers – Complete Energy Norm Analysis , Paper AIAA-2005-5314 – p. 6
Problem formulation Geometry. Flat plate and sphere. Zuccher, S., Tumin, A., Reshotko, E., Optimal Disturbances in Compressible Boundary Layers – Complete Energy Norm Analysis , Paper AIAA-2005-5314 – p. 7
Problem formulation Geometry. Flat plate and sphere. Regimes. Compressible, sub/supersonic. Possibly reducing to incompressible regime for M → 0 . Zuccher, S., Tumin, A., Reshotko, E., Optimal Disturbances in Compressible Boundary Layers – Complete Energy Norm Analysis , Paper AIAA-2005-5314 – p. 7
Problem formulation Geometry. Flat plate and sphere. Regimes. Compressible, sub/supersonic. Possibly reducing to incompressible regime for M → 0 . Equations. Linearized, steady Navier-Stokes equations. Zuccher, S., Tumin, A., Reshotko, E., Optimal Disturbances in Compressible Boundary Layers – Complete Energy Norm Analysis , Paper AIAA-2005-5314 – p. 7
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Scaling (1/2) L ref is a typical scale of the geometry ( L for flat plate, R for sphere, etc.) Zuccher, S., Tumin, A., Reshotko, E., Optimal Disturbances in Compressible Boundary Layers – Complete Energy Norm Analysis , Paper AIAA-2005-5314 – p. 8
Scaling (1/2) L ref is a typical scale of the geometry ( L for flat plate, R for sphere, etc.) � H ref = ν ref L ref /U ref is a typical boundary-layer scale in the wall-normal direction Zuccher, S., Tumin, A., Reshotko, E., Optimal Disturbances in Compressible Boundary Layers – Complete Energy Norm Analysis , Paper AIAA-2005-5314 – p. 8
Scaling (1/2) L ref is a typical scale of the geometry ( L for flat plate, R for sphere, etc.) � H ref = ν ref L ref /U ref is a typical boundary-layer scale in the wall-normal direction � Flat plate. H ref = l = ν ∞ L/U ∞ ; ∞ = freestream. Zuccher, S., Tumin, A., Reshotko, E., Optimal Disturbances in Compressible Boundary Layers – Complete Energy Norm Analysis , Paper AIAA-2005-5314 – p. 8
Scaling (1/2) L ref is a typical scale of the geometry ( L for flat plate, R for sphere, etc.) � H ref = ν ref L ref /U ref is a typical boundary-layer scale in the wall-normal direction � Flat plate. H ref = l = ν ∞ L/U ∞ ; ∞ = freestream. � Sphere. H ref = ν ref R/U ref ; ref = edge-conditions at x ref . Zuccher, S., Tumin, A., Reshotko, E., Optimal Disturbances in Compressible Boundary Layers – Complete Energy Norm Analysis , Paper AIAA-2005-5314 – p. 8
Scaling (1/2) L ref is a typical scale of the geometry ( L for flat plate, R for sphere, etc.) � H ref = ν ref L ref /U ref is a typical boundary-layer scale in the wall-normal direction � Flat plate. H ref = l = ν ∞ L/U ∞ ; ∞ = freestream. � Sphere. H ref = ν ref R/U ref ; ref = edge-conditions at x ref . ǫ = H ref /L ref is a small parameter. Zuccher, S., Tumin, A., Reshotko, E., Optimal Disturbances in Compressible Boundary Layers – Complete Energy Norm Analysis , Paper AIAA-2005-5314 – p. 8
Scaling (1/2) L ref is a typical scale of the geometry ( L for flat plate, R for sphere, etc.) � H ref = ν ref L ref /U ref is a typical boundary-layer scale in the wall-normal direction � Flat plate. H ref = l = ν ∞ L/U ∞ ; ∞ = freestream. � Sphere. H ref = ν ref R/U ref ; ref = edge-conditions at x ref . ǫ = H ref /L ref is a small parameter. Flat plate. ǫ = Re − 1 / 2 , Re L = U ∞ L/ν ∞ . L Zuccher, S., Tumin, A., Reshotko, E., Optimal Disturbances in Compressible Boundary Layers – Complete Energy Norm Analysis , Paper AIAA-2005-5314 – p. 8
Scaling (1/2) L ref is a typical scale of the geometry ( L for flat plate, R for sphere, etc.) � H ref = ν ref L ref /U ref is a typical boundary-layer scale in the wall-normal direction � Flat plate. H ref = l = ν ∞ L/U ∞ ; ∞ = freestream. � Sphere. H ref = ν ref R/U ref ; ref = edge-conditions at x ref . ǫ = H ref /L ref is a small parameter. Flat plate. ǫ = Re − 1 / 2 , Re L = U ∞ L/ν ∞ . L Sphere. ǫ = Re − 1 / 2 , Re ref = U ref R/ν ref . ref Zuccher, S., Tumin, A., Reshotko, E., Optimal Disturbances in Compressible Boundary Layers – Complete Energy Norm Analysis , Paper AIAA-2005-5314 – p. 8
Scaling (2/2) From previous works, disturbance expected as streamwise vortices. The natural scaling is therefore Zuccher, S., Tumin, A., Reshotko, E., Optimal Disturbances in Compressible Boundary Layers – Complete Energy Norm Analysis , Paper AIAA-2005-5314 – p. 9
Scaling (2/2) From previous works, disturbance expected as streamwise vortices. The natural scaling is therefore x normalized with L ref , y and z scaled with ǫL ref . Zuccher, S., Tumin, A., Reshotko, E., Optimal Disturbances in Compressible Boundary Layers – Complete Energy Norm Analysis , Paper AIAA-2005-5314 – p. 9
Scaling (2/2) From previous works, disturbance expected as streamwise vortices. The natural scaling is therefore x normalized with L ref , y and z scaled with ǫL ref . u is scaled with U ref , v and w with ǫU ref . Zuccher, S., Tumin, A., Reshotko, E., Optimal Disturbances in Compressible Boundary Layers – Complete Energy Norm Analysis , Paper AIAA-2005-5314 – p. 9
Scaling (2/2) From previous works, disturbance expected as streamwise vortices. The natural scaling is therefore x normalized with L ref , y and z scaled with ǫL ref . u is scaled with U ref , v and w with ǫU ref . T with T ref and p with ǫ 2 ρ ref U 2 ref . ρ eliminated through the state equation. Zuccher, S., Tumin, A., Reshotko, E., Optimal Disturbances in Compressible Boundary Layers – Complete Energy Norm Analysis , Paper AIAA-2005-5314 – p. 9
Scaling (2/2) From previous works, disturbance expected as streamwise vortices. The natural scaling is therefore x normalized with L ref , y and z scaled with ǫL ref . u is scaled with U ref , v and w with ǫU ref . T with T ref and p with ǫ 2 ρ ref U 2 ref . ρ eliminated through the state equation. Due to the scaling, ( · ) xx << 1 . The equations are parabolic! Zuccher, S., Tumin, A., Reshotko, E., Optimal Disturbances in Compressible Boundary Layers – Complete Energy Norm Analysis , Paper AIAA-2005-5314 – p. 9
Scaling (2/2) From previous works, disturbance expected as streamwise vortices. The natural scaling is therefore x normalized with L ref , y and z scaled with ǫL ref . u is scaled with U ref , v and w with ǫU ref . T with T ref and p with ǫ 2 ρ ref U 2 ref . ρ eliminated through the state equation. Due to the scaling, ( · ) xx << 1 . The equations are parabolic! By assuming perturbations in the form q ( x, y ) exp(i βz ) (flat plate – β spanwise wavenumber) and q ( x, y ) exp(i mφ ) (sphere – m azimuthal index)... Zuccher, S., Tumin, A., Reshotko, E., Optimal Disturbances in Compressible Boundary Layers – Complete Energy Norm Analysis , Paper AIAA-2005-5314 – p. 9
Governing equations ( A f ) x = ( D f y ) x + B 0 f + B 1 f y + B 2 f yy f = [ u, v, w, T, p ] T ; A , B 0 , B 1 , B 2 , D 5 × 5 real matrices. Zuccher, S., Tumin, A., Reshotko, E., Optimal Disturbances in Compressible Boundary Layers – Complete Energy Norm Analysis , Paper AIAA-2005-5314 – p. 10
Governing equations ( A f ) x = ( D f y ) x + B 0 f + B 1 f y + B 2 f yy f = [ u, v, w, T, p ] T ; A , B 0 , B 1 , B 2 , D 5 × 5 real matrices. Boundary conditions y = 0 : u = 0; v = 0; w = 0; T = 0 y → ∞ : u → 0; w → 0; p → 0; T → 0 Zuccher, S., Tumin, A., Reshotko, E., Optimal Disturbances in Compressible Boundary Layers – Complete Energy Norm Analysis , Paper AIAA-2005-5314 – p. 10
Governing equations ( A f ) x = ( D f y ) x + B 0 f + B 1 f y + B 2 f yy f = [ u, v, w, T, p ] T ; A , B 0 , B 1 , B 2 , D 5 × 5 real matrices. Boundary conditions y = 0 : u = 0; v = 0; w = 0; T = 0 y → ∞ : u → 0; w → 0; p → 0; T → 0 More compactly ( H 1 f ) x + H 2 f = 0 with H 1 = A − D ( · ) y ; H 2 = − B 0 − B 1 ( · ) y − B 2 ( · ) yy Zuccher, S., Tumin, A., Reshotko, E., Optimal Disturbances in Compressible Boundary Layers – Complete Energy Norm Analysis , Paper AIAA-2005-5314 – p. 10
Objective function (1/2) Caveat ! Results depend on the choice of the objective function. Zuccher, S., Tumin, A., Reshotko, E., Optimal Disturbances in Compressible Boundary Layers – Complete Energy Norm Analysis , Paper AIAA-2005-5314 – p. 11
Objective function (1/2) Caveat ! Results depend on the choice of the objective function. Physics dominated by streamwise vortices. Zuccher, S., Tumin, A., Reshotko, E., Optimal Disturbances in Compressible Boundary Layers – Complete Energy Norm Analysis , Paper AIAA-2005-5314 – p. 11
Objective function (1/2) Caveat ! Results depend on the choice of the objective function. Physics dominated by streamwise vortices. Common choices of the energy norms. Zuccher, S., Tumin, A., Reshotko, E., Optimal Disturbances in Compressible Boundary Layers – Complete Energy Norm Analysis , Paper AIAA-2005-5314 – p. 11
Objective function (1/2) Caveat ! Results depend on the choice of the objective function. Physics dominated by streamwise vortices. Common choices of the energy norms. Inlet. v in � = 0 and w in � = 0 ( u in = T in = 0 ). Zuccher, S., Tumin, A., Reshotko, E., Optimal Disturbances in Compressible Boundary Layers – Complete Energy Norm Analysis , Paper AIAA-2005-5314 – p. 11
Objective function (1/2) Caveat ! Results depend on the choice of the objective function. Physics dominated by streamwise vortices. Common choices of the energy norms. Inlet. v in � = 0 and w in � = 0 ( u in = T in = 0 ). Outlet. v out = 0 and w out = 0 ( u in � = 0; T in � = 0 ). Zuccher, S., Tumin, A., Reshotko, E., Optimal Disturbances in Compressible Boundary Layers – Complete Energy Norm Analysis , Paper AIAA-2005-5314 – p. 11
Objective function (1/2) Caveat ! Results depend on the choice of the objective function. Physics dominated by streamwise vortices. Common choices of the energy norms. Inlet. v in � = 0 and w in � = 0 ( u in = T in = 0 ). Outlet. v out = 0 and w out = 0 ( u in � = 0; T in � = 0 ). Blunt body. Largest transient growth close to the stagnation point. Zuccher, S., Tumin, A., Reshotko, E., Optimal Disturbances in Compressible Boundary Layers – Complete Energy Norm Analysis , Paper AIAA-2005-5314 – p. 11
Objective function (1/2) Caveat ! Results depend on the choice of the objective function. Physics dominated by streamwise vortices. Common choices of the energy norms. Inlet. v in � = 0 and w in � = 0 ( u in = T in = 0 ). Outlet. v out = 0 and w out = 0 ( u in � = 0; T in � = 0 ). Blunt body. Largest transient growth close to the stagnation point. Due to short x -interval, a streaks-dominated flow field might not be completely established. Zuccher, S., Tumin, A., Reshotko, E., Optimal Disturbances in Compressible Boundary Layers – Complete Energy Norm Analysis , Paper AIAA-2005-5314 – p. 11
Objective function (1/2) Caveat ! Results depend on the choice of the objective function. Physics dominated by streamwise vortices. Common choices of the energy norms. Inlet. v in � = 0 and w in � = 0 ( u in = T in = 0 ). Outlet. v out = 0 and w out = 0 ( u in � = 0; T in � = 0 ). Blunt body. Largest transient growth close to the stagnation point. Due to short x -interval, a streaks-dominated flow field might not be completely established. Contribution of v out and w out could be non negligible. Zuccher, S., Tumin, A., Reshotko, E., Optimal Disturbances in Compressible Boundary Layers – Complete Energy Norm Analysis , Paper AIAA-2005-5314 – p. 11
Objective function (1/2) Caveat ! Results depend on the choice of the objective function. Physics dominated by streamwise vortices. Common choices of the energy norms. Inlet. v in � = 0 and w in � = 0 ( u in = T in = 0 ). Outlet. v out = 0 and w out = 0 ( u in � = 0; T in � = 0 ). Blunt body. Largest transient growth close to the stagnation point. Due to short x -interval, a streaks-dominated flow field might not be completely established. Contribution of v out and w out could be non negligible. Outlet norm. FEN vs. PEN Zuccher, S., Tumin, A., Reshotko, E., Optimal Disturbances in Compressible Boundary Layers – Complete Energy Norm Analysis , Paper AIAA-2005-5314 – p. 11
Objective function (2/2) Mack’s energy norm (derived for flat plate and temporal problem), after scaling and using state equation, � ∞ p s out T 2 � � ρ s out ( u 2 out + v 2 out + w 2 out E out = out ) + dy ( γ − 1) T s 2 out M 2 0 Zuccher, S., Tumin, A., Reshotko, E., Optimal Disturbances in Compressible Boundary Layers – Complete Energy Norm Analysis , Paper AIAA-2005-5314 – p. 12
Objective function (2/2) Mack’s energy norm (derived for flat plate and temporal problem), after scaling and using state equation, � ∞ p s out T 2 � � ρ s out ( u 2 out + v 2 out + w 2 out E out = out ) + dy ( γ − 1) T s 2 out M 2 0 � ∞ � � out � f T or in matrix form as E out = dy , with M out f out 0 � � p s out � M out = diag ρ s out , ρ s out , ρ s out , out M 2 , 0 . ( γ − 1) T s 2 Zuccher, S., Tumin, A., Reshotko, E., Optimal Disturbances in Compressible Boundary Layers – Complete Energy Norm Analysis , Paper AIAA-2005-5314 – p. 12
Objective function (2/2) Mack’s energy norm (derived for flat plate and temporal problem), after scaling and using state equation, � ∞ p s out T 2 � � ρ s out ( u 2 out + v 2 out + w 2 out E out = out ) + dy ( γ − 1) T s 2 out M 2 0 � ∞ � � out � f T or in matrix form as E out = dy , with M out f out 0 � � p s out � M out = diag ρ s out , ρ s out , ρ s out , out M 2 , 0 . ( γ − 1) T s 2 Initial energy of the perturbation � ∞ � ∞ � � � � in � ρ s in ( v 2 in + w 2 f T E in = in ) dy ⇒ E in = dy M in f in 0 0 Zuccher, S., Tumin, A., Reshotko, E., Optimal Disturbances in Compressible Boundary Layers – Complete Energy Norm Analysis , Paper AIAA-2005-5314 – p. 12
Constrained optimization (1/3) Our constraints are the governing equations, boundary conditions and the normalization condition E in = E 0 . Zuccher, S., Tumin, A., Reshotko, E., Optimal Disturbances in Compressible Boundary Layers – Complete Energy Norm Analysis , Paper AIAA-2005-5314 – p. 13
Constrained optimization (1/3) Our constraints are the governing equations, boundary conditions and the normalization condition E in = E 0 . After discretization ( M 0 ⇔ � M in and M N ⇔ � M out ), Zuccher, S., Tumin, A., Reshotko, E., Optimal Disturbances in Compressible Boundary Layers – Complete Energy Norm Analysis , Paper AIAA-2005-5314 – p. 13
Constrained optimization (1/3) Our constraints are the governing equations, boundary conditions and the normalization condition E in = E 0 . After discretization ( M 0 ⇔ � M in and M N ⇔ � M out ), objective function J = f T N M N f N Zuccher, S., Tumin, A., Reshotko, E., Optimal Disturbances in Compressible Boundary Layers – Complete Energy Norm Analysis , Paper AIAA-2005-5314 – p. 13
Constrained optimization (1/3) Our constraints are the governing equations, boundary conditions and the normalization condition E in = E 0 . After discretization ( M 0 ⇔ � M in and M N ⇔ � M out ), objective function J = f T N M N f N constraint E in = E 0 ⇒ f T 0 M 0 f 0 = E 0 Zuccher, S., Tumin, A., Reshotko, E., Optimal Disturbances in Compressible Boundary Layers – Complete Energy Norm Analysis , Paper AIAA-2005-5314 – p. 13
Constrained optimization (1/3) Our constraints are the governing equations, boundary conditions and the normalization condition E in = E 0 . After discretization ( M 0 ⇔ � M in and M N ⇔ � M out ), objective function J = f T N M N f N constraint E in = E 0 ⇒ f T 0 M 0 f 0 = E 0 governing equations (BC included) C n +1 f n +1 = B n f n Zuccher, S., Tumin, A., Reshotko, E., Optimal Disturbances in Compressible Boundary Layers – Complete Energy Norm Analysis , Paper AIAA-2005-5314 – p. 13
Constrained optimization (1/3) Our constraints are the governing equations, boundary conditions and the normalization condition E in = E 0 . After discretization ( M 0 ⇔ � M in and M N ⇔ � M out ), objective function J = f T N M N f N constraint E in = E 0 ⇒ f T 0 M 0 f 0 = E 0 governing equations (BC included) C n +1 f n +1 = B n f n The augmented functional L is N − 1 � � � L ( f 0 , . . . , f N ) = f T N M N f N + λ 0 [ f T p T 0 M 0 f 0 − E 0 ] + n ( C n +1 f n +1 − B n f n ) n =0 Zuccher, S., Tumin, A., Reshotko, E., Optimal Disturbances in Compressible Boundary Layers – Complete Energy Norm Analysis , Paper AIAA-2005-5314 – p. 13
Constrained optimization (1/3) Our constraints are the governing equations, boundary conditions and the normalization condition E in = E 0 . After discretization ( M 0 ⇔ � M in and M N ⇔ � M out ), objective function J = f T N M N f N constraint E in = E 0 ⇒ f T 0 M 0 f 0 = E 0 governing equations (BC included) C n +1 f n +1 = B n f n The augmented functional L is N − 1 � � � L ( f 0 , . . . , f N ) = f T N M N f N + λ 0 [ f T p T 0 M 0 f 0 − E 0 ] + n ( C n +1 f n +1 − B n f n ) n =0 Zuccher, S., Tumin, A., Reshotko, E., Optimal Disturbances in Compressible Boundary Layers – Complete Energy Norm Analysis , Paper AIAA-2005-5314 – p. 13
Constrained optimization (1/3) Our constraints are the governing equations, boundary conditions and the normalization condition E in = E 0 . After discretization ( M 0 ⇔ � M in and M N ⇔ � M out ), objective function J = f T N M N f N constraint E in = E 0 ⇒ f T 0 M 0 f 0 = E 0 governing equations (BC included) C n +1 f n +1 = B n f n The augmented functional L is N − 1 � � � L ( f 0 , . . . , f N ) = f T N M N f N + λ 0 [ f T p T 0 M 0 f 0 − E 0 ] + n ( C n +1 f n +1 − B n f n ) n =0 Zuccher, S., Tumin, A., Reshotko, E., Optimal Disturbances in Compressible Boundary Layers – Complete Energy Norm Analysis , Paper AIAA-2005-5314 – p. 13
Constrained optimization (1/3) Our constraints are the governing equations, boundary conditions and the normalization condition E in = E 0 . After discretization ( M 0 ⇔ � M in and M N ⇔ � M out ), objective function J = f T N M N f N constraint E in = E 0 ⇒ f T 0 M 0 f 0 = E 0 governing equations (BC included) C n +1 f n +1 = B n f n The augmented functional L is N − 1 � � � L ( f 0 , . . . , f N ) = f T N M N f N + λ 0 [ f T p T 0 M 0 f 0 − E 0 ] + n ( C n +1 f n +1 − B n f n ) n =0 Zuccher, S., Tumin, A., Reshotko, E., Optimal Disturbances in Compressible Boundary Layers – Complete Energy Norm Analysis , Paper AIAA-2005-5314 – p. 13
Constrained optimization (1/3) Our constraints are the governing equations, boundary conditions and the normalization condition E in = E 0 . After discretization ( M 0 ⇔ � M in and M N ⇔ � M out ), objective function J = f T N M N f N constraint E in = E 0 ⇒ f T 0 M 0 f 0 = E 0 governing equations (BC included) C n +1 f n +1 = B n f n The augmented functional L is N − 1 � � � L ( f 0 , . . . , f N ) = f T N M N f N + λ 0 [ f T p T 0 M 0 f 0 − E 0 ] + n ( C n +1 f n +1 − B n f n ) n =0 with λ 0 and (vector) p n Lagrangian multipliers. Zuccher, S., Tumin, A., Reshotko, E., Optimal Disturbances in Compressible Boundary Layers – Complete Energy Norm Analysis , Paper AIAA-2005-5314 – p. 13
Constrained optimization (2/3) By adding and subtracting p T n +1 B n +1 f n +1 in the summation, � � � � N − 1 N − 1 � � p T p T n C n +1 f n +1 − p T n ( C n +1 f n +1 − B n f n ) = + n +1 B n +1 f n +1 n =0 n =0 N − 1 � � � p T n +1 B n +1 f n +1 − p T n B n f n n =0 � � N − 1 � p T n C n +1 f n +1 − p T = + n +1 B n +1 f n +1 n =0 p T N B N f N − p T 0 B 0 f 0 , Zuccher, S., Tumin, A., Reshotko, E., Optimal Disturbances in Compressible Boundary Layers – Complete Energy Norm Analysis , Paper AIAA-2005-5314 – p. 14
Constrained optimization (2/3) By adding and subtracting p T n +1 B n +1 f n +1 in the summation, � � � � N − 1 N − 1 � � p T p T n C n +1 f n +1 − p T n ( C n +1 f n +1 − B n f n ) = + n +1 B n +1 f n +1 n =0 n =0 N − 1 � � � p T n +1 B n +1 f n +1 − p T n B n f n n =0 � � N − 1 � p T n C n +1 f n +1 − p T = + n +1 B n +1 f n +1 n =0 p T N B N f N − p T 0 B 0 f 0 , � � N − 1 � p T n C n +1 f n +1 − p T f T L ( f 0 , . . . , f N ) = N M N f N + + n +1 B n +1 f n +1 n =0 p T N B N f N − p T 0 B 0 f 0 + λ 0 [ f T 0 M 0 f 0 − E 0 ] . Zuccher, S., Tumin, A., Reshotko, E., Optimal Disturbances in Compressible Boundary Layers – Complete Energy Norm Analysis , Paper AIAA-2005-5314 – p. 14
Constrained optimization (2/3) By adding and subtracting p T n +1 B n +1 f n +1 in the summation, � � � � N − 1 N − 1 � � p T p T n C n +1 f n +1 − p T n ( C n +1 f n +1 − B n f n ) = + n +1 B n +1 f n +1 n =0 n =0 N − 1 � � � p T n +1 B n +1 f n +1 − p T n B n f n n =0 � � N − 1 � p T n C n +1 f n +1 − p T = + n +1 B n +1 f n +1 n =0 p T N B N f N − p T 0 B 0 f 0 , � � N − 1 � p T n C n +1 f n +1 − p T f T L ( f 0 , . . . , f N ) = N M N f N + + n +1 B n +1 f n +1 n =0 p T N B N f N − p T 0 B 0 f 0 + λ 0 [ f T 0 M 0 f 0 − E 0 ] . Stationary condition � � N − 2 � δ L = 0 ⇒ δ L δ L + δ L δ f 0 + δ f n +1 δ f N = 0 δ f 0 δ f n +1 δ f N n =0 Zuccher, S., Tumin, A., Reshotko, E., Optimal Disturbances in Compressible Boundary Layers – Complete Energy Norm Analysis , Paper AIAA-2005-5314 – p. 14
Constrained optimization (3/3) δ L − p T 0 B 0 + 2 λ 0 f T = 0 M 0 = 0 δ f 0 δ L p T n C n +1 − p T = n +1 B n +1 = 0 , n = 0 , . . . , N − 2 δ f n +1 δ L 2 f T N M N + p T = N B N = 0 δ f N Zuccher, S., Tumin, A., Reshotko, E., Optimal Disturbances in Compressible Boundary Layers – Complete Energy Norm Analysis , Paper AIAA-2005-5314 – p. 15
Constrained optimization (3/3) δ L − p T 0 B 0 + 2 λ 0 f T = 0 M 0 = 0 δ f 0 δ L p T n C n +1 − p T = n +1 B n +1 = 0 , n = 0 , . . . , N − 2 δ f n +1 δ L 2 f T N M N + p T = N B N = 0 δ f N ( p T 0 B 0 ) j M 0 jj � = 0 if 2 λ 0 M 0 jj Inlet conditions : f 0 j = 0 M 0 jj = 0 if p T n C n +1 − p T “Adjoint” equations : n +1 B n +1 = 0 B T N p N = − 2 M T Oulet conditions : N f N Zuccher, S., Tumin, A., Reshotko, E., Optimal Disturbances in Compressible Boundary Layers – Complete Energy Norm Analysis , Paper AIAA-2005-5314 – p. 15
Constrained optimization (3/3) δ L − p T 0 B 0 + 2 λ 0 f T = 0 M 0 = 0 δ f 0 δ L p T n C n +1 − p T = n +1 B n +1 = 0 , n = 0 , . . . , N − 2 δ f n +1 δ L 2 f T N M N + p T = N B N = 0 δ f N ( p T 0 B 0 ) j M 0 jj � = 0 if 2 λ 0 M 0 jj Inlet conditions : f 0 j = 0 M 0 jj = 0 if p T n C n +1 − p T “Adjoint” equations : n +1 B n +1 = 0 B T N p N = − 2 M T Oulet conditions : N f N Zuccher, S., Tumin, A., Reshotko, E., Optimal Disturbances in Compressible Boundary Layers – Complete Energy Norm Analysis , Paper AIAA-2005-5314 – p. 15
Constrained optimization (3/3) δ L − p T 0 B 0 + 2 λ 0 f T = 0 M 0 = 0 δ f 0 δ L p T n C n +1 − p T = n +1 B n +1 = 0 , n = 0 , . . . , N − 2 δ f n +1 δ L 2 f T N M N + p T = N B N = 0 δ f N ( p T 0 B 0 ) j M 0 jj � = 0 if 2 λ 0 M 0 jj Inlet conditions : f 0 j = 0 M 0 jj = 0 if p T n C n +1 − p T “Adjoint” equations : n +1 B n +1 = 0 B T N p N = − 2 M T Oulet conditions : N f N Zuccher, S., Tumin, A., Reshotko, E., Optimal Disturbances in Compressible Boundary Layers – Complete Energy Norm Analysis , Paper AIAA-2005-5314 – p. 15
An optimization algorithm 1. guessed initial condition f (0) in Zuccher, S., Tumin, A., Reshotko, E., Optimal Disturbances in Compressible Boundary Layers – Complete Energy Norm Analysis , Paper AIAA-2005-5314 – p. 16
An optimization algorithm 1. guessed initial condition f (0) in 2. solution of forward problem with the IC f ( n ) in Zuccher, S., Tumin, A., Reshotko, E., Optimal Disturbances in Compressible Boundary Layers – Complete Energy Norm Analysis , Paper AIAA-2005-5314 – p. 16
An optimization algorithm 1. guessed initial condition f (0) in 2. solution of forward problem with the IC f ( n ) in 3. evaluation of objective function J ( n ) = E ( n ) out . If |J ( n ) / J ( n − 1) − 1 | < ǫ t optimization converged Zuccher, S., Tumin, A., Reshotko, E., Optimal Disturbances in Compressible Boundary Layers – Complete Energy Norm Analysis , Paper AIAA-2005-5314 – p. 16
An optimization algorithm 1. guessed initial condition f (0) in 2. solution of forward problem with the IC f ( n ) in 3. evaluation of objective function J ( n ) = E ( n ) out . If |J ( n ) / J ( n − 1) − 1 | < ǫ t optimization converged 4. if |J ( n ) / J ( n − 1) − 1 | > ǫ t outlet conditions provide the “initial” conditions for the backward problem at x = x out Zuccher, S., Tumin, A., Reshotko, E., Optimal Disturbances in Compressible Boundary Layers – Complete Energy Norm Analysis , Paper AIAA-2005-5314 – p. 16
An optimization algorithm 1. guessed initial condition f (0) in 2. solution of forward problem with the IC f ( n ) in 3. evaluation of objective function J ( n ) = E ( n ) out . If |J ( n ) / J ( n − 1) − 1 | < ǫ t optimization converged 4. if |J ( n ) / J ( n − 1) − 1 | > ǫ t outlet conditions provide the “initial” conditions for the backward problem at x = x out 5. backward solution of the “adjoint” problem from x = x out to x = x in Zuccher, S., Tumin, A., Reshotko, E., Optimal Disturbances in Compressible Boundary Layers – Complete Energy Norm Analysis , Paper AIAA-2005-5314 – p. 16
An optimization algorithm 1. guessed initial condition f (0) in 2. solution of forward problem with the IC f ( n ) in 3. evaluation of objective function J ( n ) = E ( n ) out . If |J ( n ) / J ( n − 1) − 1 | < ǫ t optimization converged 4. if |J ( n ) / J ( n − 1) − 1 | > ǫ t outlet conditions provide the “initial” conditions for the backward problem at x = x out 5. backward solution of the “adjoint” problem from x = x out to x = x in 6. from the inlet conditions, update of the initial condition for the forward problem f ( n +1) in Zuccher, S., Tumin, A., Reshotko, E., Optimal Disturbances in Compressible Boundary Layers – Complete Energy Norm Analysis , Paper AIAA-2005-5314 – p. 16
An optimization algorithm 1. guessed initial condition f (0) in 2. solution of forward problem with the IC f ( n ) in 3. evaluation of objective function J ( n ) = E ( n ) out . If |J ( n ) / J ( n − 1) − 1 | < ǫ t optimization converged 4. if |J ( n ) / J ( n − 1) − 1 | > ǫ t outlet conditions provide the “initial” conditions for the backward problem at x = x out 5. backward solution of the “adjoint” problem from x = x out to x = x in 6. from the inlet conditions, update of the initial condition for the forward problem f ( n +1) in 7. repeat from step 2 on Zuccher, S., Tumin, A., Reshotko, E., Optimal Disturbances in Compressible Boundary Layers – Complete Energy Norm Analysis , Paper AIAA-2005-5314 – p. 16
Results Discretization. 2nd-order backward finite differences in x and 4th-order finite differences in y . Zuccher, S., Tumin, A., Reshotko, E., Optimal Disturbances in Compressible Boundary Layers – Complete Energy Norm Analysis , Paper AIAA-2005-5314 – p. 17
Results Discretization. 2nd-order backward finite differences in x and 4th-order finite differences in y . Uneven grids in both x and y . Zuccher, S., Tumin, A., Reshotko, E., Optimal Disturbances in Compressible Boundary Layers – Complete Energy Norm Analysis , Paper AIAA-2005-5314 – p. 17
Results Discretization. 2nd-order backward finite differences in x and 4th-order finite differences in y . Uneven grids in both x and y . Code verified against results by Tumin & Reshotko (2003, 2004) obtained with spectral collocation method. Zuccher, S., Tumin, A., Reshotko, E., Optimal Disturbances in Compressible Boundary Layers – Complete Energy Norm Analysis , Paper AIAA-2005-5314 – p. 17
Results Discretization. 2nd-order backward finite differences in x and 4th-order finite differences in y . Uneven grids in both x and y . Code verified against results by Tumin & Reshotko (2003, 2004) obtained with spectral collocation method. Inlet norm includes v in and w in only. Zuccher, S., Tumin, A., Reshotko, E., Optimal Disturbances in Compressible Boundary Layers – Complete Energy Norm Analysis , Paper AIAA-2005-5314 – p. 17
Results Discretization. 2nd-order backward finite differences in x and 4th-order finite differences in y . Uneven grids in both x and y . Code verified against results by Tumin & Reshotko (2003, 2004) obtained with spectral collocation method. Inlet norm includes v in and w in only. Outlet norm. Partial Energy Norm (PEN) u out and T out only. Zuccher, S., Tumin, A., Reshotko, E., Optimal Disturbances in Compressible Boundary Layers – Complete Energy Norm Analysis , Paper AIAA-2005-5314 – p. 17
Results Discretization. 2nd-order backward finite differences in x and 4th-order finite differences in y . Uneven grids in both x and y . Code verified against results by Tumin & Reshotko (2003, 2004) obtained with spectral collocation method. Inlet norm includes v in and w in only. Outlet norm. Partial Energy Norm (PEN) u out and T out only. Full Energy Norm (FEN) u out , v out , w out , T out . Zuccher, S., Tumin, A., Reshotko, E., Optimal Disturbances in Compressible Boundary Layers – Complete Energy Norm Analysis , Paper AIAA-2005-5314 – p. 17
Results Discretization. 2nd-order backward finite differences in x and 4th-order finite differences in y . Uneven grids in both x and y . Code verified against results by Tumin & Reshotko (2003, 2004) obtained with spectral collocation method. Inlet norm includes v in and w in only. Outlet norm. Partial Energy Norm (PEN) u out and T out only. Full Energy Norm (FEN) u out , v out , w out , T out . FEN depends on Re , PEN is Re -independent. Zuccher, S., Tumin, A., Reshotko, E., Optimal Disturbances in Compressible Boundary Layers – Complete Energy Norm Analysis , Paper AIAA-2005-5314 – p. 17
Results – Flat plate 0.0018 Re → ∞ 0.0016 Re = 10 4 Re = 10 3 0.0014 0.0012 G/Re 0.001 0.0008 0.0006 0.0004 0.0002 0 0.2 0.4 0.6 0.8 1 β Objective function G/Re : effect of Re and β for M = 3 , T w /T ad = 1 , x in = 0 x out = 1 . 0 , FEN. ⇒ Reynolds number effects only for Re < 10 4 . Zuccher, S., Tumin, A., Reshotko, E., Optimal Disturbances in Compressible Boundary Layers – Complete Energy Norm Analysis , Paper AIAA-2005-5314 – p. 18
Results – Flat plate 0.02 0.018 0.016 0.014 0.012 G/Re 0.01 0.008 0.006 0.004 0.002 0 0.2 0.4 0.6 0.8 1 β Objective function G/Re : effect of β , T w /T ad and norm choice (PEN vs. FEN) for M = 0 . 5 , Re = 10 3 , x in = 0 x out = 1 . 0 . ✷ , T w /T ad = 1 . 00 ; ◦ , T w /T ad = 0 . 50 ; △ , T w /T ad = 0 . 25 . ⇒ No remarkable norm effects; cold wall destabilizing factor. Zuccher, S., Tumin, A., Reshotko, E., Optimal Disturbances in Compressible Boundary Layers – Complete Energy Norm Analysis , Paper AIAA-2005-5314 – p. 19
Results – Flat plate 0.0035 0.003 0.0025 0.002 G/Re 0.0015 0.001 0.0005 0 0.2 0.4 0.6 0.8 1 β Objective function G/Re : effect of β , T w /T ad and norm choice (PEN vs. FEN) for M = 1 . 5 , Re = 10 3 , x in = 0 x out = 1 . 0 . ✷ , T w /T ad = 1 . 00 ; ◦ , T w /T ad = 0 . 50 ; △ , T w /T ad = 0 . 25 . ⇒ Shift of the curves maximum, enhanced difference between norms ( T w /T ad = 1 . 00 ). Zuccher, S., Tumin, A., Reshotko, E., Optimal Disturbances in Compressible Boundary Layers – Complete Energy Norm Analysis , Paper AIAA-2005-5314 – p. 20
Results – Flat plate 0.0018 0.0016 0.0014 0.0012 G/Re 0.001 0.0008 0.0006 0.0004 0.0002 0 0 0.2 0.4 0.6 0.8 1 β Objective function G/Re : effect of β , T w /T ad and norm choice (PEN vs. FEN) for M = 3 , Re = 10 3 , x in = 0 x out = 1 . 0 . ✷ , T w /T ad = 1 . 00 ; ◦ , T w /T ad = 0 . 50 ; △ , T w /T ad = 0 . 25 . ⇒ Up to 17% difference for low values of β . Zuccher, S., Tumin, A., Reshotko, E., Optimal Disturbances in Compressible Boundary Layers – Complete Energy Norm Analysis , Paper AIAA-2005-5314 – p. 21
Results – Flat plate 0.0025 0.002 0.0015 G/Re 0.001 0.0005 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 β Objective function G/Re : effect of x in and β and norm choice (PEN vs. FEN) for M = 3 , T w /T ad = 1 , x out = 1 . 0 . ✷ , x in = 0 . 0 ; ◦ , x in = 0 . 2 ; △ , x in = 0 . 4 . ⇒ Up to 60% difference for x in = 0 . 4 and β = 0 . 1 . Zuccher, S., Tumin, A., Reshotko, E., Optimal Disturbances in Compressible Boundary Layers – Complete Energy Norm Analysis , Paper AIAA-2005-5314 – p. 22
Results – Flat plate 0.4 0.2 PEN PEN 0.15 0.3 v v FEN FEN 0.1 0.2 v out ( y ) , w out ( y ) v in ( y ) , w in ( y ) 0.05 0.1 0 0 -0.05 -0.1 -0.1 -0.2 -0.15 w w -0.3 -0.2 -0.4 -0.25 0 5 10 15 20 0 5 10 15 20 y y Inlet and outlet profiles: effect of norm choice (PEN vs. FEN) for M = 3 . 0 , Re = 10 3 , x in = 0 . 4 , x out = 1 . 0 and β = 0 . 1 . ⇒ No significant changes in v in , some discrepancies in w in ; larger effects on v out , rather than on w out . No significant effects on u out and T out . Zuccher, S., Tumin, A., Reshotko, E., Optimal Disturbances in Compressible Boundary Layers – Complete Energy Norm Analysis , Paper AIAA-2005-5314 – p. 23
Results – Sphere 0.0014 Tumin & Reshotko (2004) θ ∈ [2; 5] 0.0012 θ ∈ [3; 5] 0.001 θ ∈ [5; 10] θ ∈ [10; 15] θ ∈ [15; 20] 0.0008 Gǫ 2 θ ∈ [25; 30] 0.0006 0.0004 0.0002 0 0 0.2 0.4 0.6 0.8 1 m � Objective function Gǫ 2 : effect of interval location and � m = mǫ for θ ref = 30 . 0 deg, T w /T ad = 0 . 5 , ǫ = 10 − 3 . PEN. ⇒ Largest gain for small θ out − θ in ; strongest transient growth close to the stagnation point. Zuccher, S., Tumin, A., Reshotko, E., Optimal Disturbances in Compressible Boundary Layers – Complete Energy Norm Analysis , Paper AIAA-2005-5314 – p. 24
Results – Sphere 0.0012 0.0011 0.001 0.0009 Gǫ 2 0.0008 0.0007 0.0006 0.0005 0.0004 0.02 0.04 0.06 0.08 0.1 m � Objective function Gǫ 2 : effect of ǫ , energy norm (PEN vs. FEN) and � m = mǫ for θ in = 2 . 0 deg, θ out = 5 . 0 deg, θ ref = 30 . 0 deg, T w /T ad = 0 . 5 . ✷ , ǫ = 1 · 10 − 3 ; ◦ , ǫ = 2 · 10 − 3 ; △ , ǫ = 3 · 10 − 3 . ⇒ Maximum appreciable difference within 1%. Effect increases with ǫ . Zuccher, S., Tumin, A., Reshotko, E., Optimal Disturbances in Compressible Boundary Layers – Complete Energy Norm Analysis , Paper AIAA-2005-5314 – p. 25
Conclusions Zuccher, S., Tumin, A., Reshotko, E., Optimal Disturbances in Compressible Boundary Layers – Complete Energy Norm Analysis , Paper AIAA-2005-5314 – p. 26
Conclusions √ Efficient and robust numerical method for computing compressible optimal perturbations on flat plate and sphere. Zuccher, S., Tumin, A., Reshotko, E., Optimal Disturbances in Compressible Boundary Layers – Complete Energy Norm Analysis , Paper AIAA-2005-5314 – p. 26
Conclusions √ Efficient and robust numerical method for computing compressible optimal perturbations on flat plate and sphere. √ Adjoint-based optimization technique in the discrete framework and automatic in/out-let conditions. Zuccher, S., Tumin, A., Reshotko, E., Optimal Disturbances in Compressible Boundary Layers – Complete Energy Norm Analysis , Paper AIAA-2005-5314 – p. 26
Conclusions √ Efficient and robust numerical method for computing compressible optimal perturbations on flat plate and sphere. √ Adjoint-based optimization technique in the discrete framework and automatic in/out-let conditions. √ Analysis including full energy norm at the outlet. Zuccher, S., Tumin, A., Reshotko, E., Optimal Disturbances in Compressible Boundary Layers – Complete Energy Norm Analysis , Paper AIAA-2005-5314 – p. 26
Conclusions √ Efficient and robust numerical method for computing compressible optimal perturbations on flat plate and sphere. √ Adjoint-based optimization technique in the discrete framework and automatic in/out-let conditions. √ Analysis including full energy norm at the outlet. √ Flat plate. For Re = 10 3 , significant difference in G/Re (up to 62%) between PEN and FEN. Effect of M and x in . No effect in subsonic basic flow. If Re > 10 4 , v out and w out do not play significant role. Zuccher, S., Tumin, A., Reshotko, E., Optimal Disturbances in Compressible Boundary Layers – Complete Energy Norm Analysis , Paper AIAA-2005-5314 – p. 26
Conclusions √ Efficient and robust numerical method for computing compressible optimal perturbations on flat plate and sphere. √ Adjoint-based optimization technique in the discrete framework and automatic in/out-let conditions. √ Analysis including full energy norm at the outlet. √ Flat plate. For Re = 10 3 , significant difference in G/Re (up to 62%) between PEN and FEN. Effect of M and x in . No effect in subsonic basic flow. If Re > 10 4 , v out and w out do not play significant role. √ Sphere. Largest Gǫ 2 close to the stagnation point and for small range of θ . No significant role played by v out and w out in the interesting range of parameters. Zuccher, S., Tumin, A., Reshotko, E., Optimal Disturbances in Compressible Boundary Layers – Complete Energy Norm Analysis , Paper AIAA-2005-5314 – p. 26
The End! Zuccher, S., Tumin, A., Reshotko, E., Optimal Disturbances in Compressible Boundary Layers – Complete Energy Norm Analysis , Paper AIAA-2005-5314 – p. 27
Are we missing something? At non-infinitesimal level of disturbance streaks are observed on a flat plate, instead of Tollmien–Schlichting waves. Zuccher, S., Tumin, A., Reshotko, E., Optimal Disturbances in Compressible Boundary Layers – Complete Energy Norm Analysis , Paper AIAA-2005-5314 – p. 28
Are we missing something? At non-infinitesimal level of disturbance streaks are observed on a flat plate, instead of Tollmien–Schlichting waves. Linear Stability Theory (classical modal approach) fails even for the simplest geometries (Hagen-Poiseuille pipe flow, predicted stability vs. Re crex ≈ 2300 )! Zuccher, S., Tumin, A., Reshotko, E., Optimal Disturbances in Compressible Boundary Layers – Complete Energy Norm Analysis , Paper AIAA-2005-5314 – p. 28
Are we missing something? At non-infinitesimal level of disturbance streaks are observed on a flat plate, instead of Tollmien–Schlichting waves. Linear Stability Theory (classical modal approach) fails even for the simplest geometries (Hagen-Poiseuille pipe flow, predicted stability vs. Re crex ≈ 2300 )! Certain transitional phenomena have no explanation yet, e.g. the “blunt body paradox” on spherical fore-bodies at super/hypersonic speeds. Zuccher, S., Tumin, A., Reshotko, E., Optimal Disturbances in Compressible Boundary Layers – Complete Energy Norm Analysis , Paper AIAA-2005-5314 – p. 28
Are we missing something? At non-infinitesimal level of disturbance streaks are observed on a flat plate, instead of Tollmien–Schlichting waves. Linear Stability Theory (classical modal approach) fails even for the simplest geometries (Hagen-Poiseuille pipe flow, predicted stability vs. Re crex ≈ 2300 )! Certain transitional phenomena have no explanation yet, e.g. the “blunt body paradox” on spherical fore-bodies at super/hypersonic speeds. There must exist another mechanism, not related to the eigenvalue analysis: transient growth. Zuccher, S., Tumin, A., Reshotko, E., Optimal Disturbances in Compressible Boundary Layers – Complete Energy Norm Analysis , Paper AIAA-2005-5314 – p. 28
Alternative paths of BL transition M. V. Morkovin, E. Reshotko, and T. Herbert, (1994),“Transition in open flow systems – A re- assessment”, Bull. Am. Phys. Soc. 39 , 1882. Zuccher, S., Tumin, A., Reshotko, E., Optimal Disturbances in Compressible Boundary Layers – Complete Energy Norm Analysis , Paper AIAA-2005-5314 – p. 29
Alternative paths of BL transition “At the present time, no mathematical model exists that can predict the transition Reynolds number on a flat plate”! Saric et al., Annu. Rev. Fluid Mech. 2002. 34 :291–319 M. V. Morkovin, E. Reshotko, and T. Herbert, (1994),“Transition in open flow systems – A re- assessment”, Bull. Am. Phys. Soc. 39 , 1882. Zuccher, S., Tumin, A., Reshotko, E., Optimal Disturbances in Compressible Boundary Layers – Complete Energy Norm Analysis , Paper AIAA-2005-5314 – p. 29
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