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Control of boundary layer flow transition via distributed

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Abstract

A reduced-order linear feedback controller, which is used to control the linear disturbance in two-dimensional plane Poiseuille flow, is applied to a boundary layer flow for stability control. Using model reduction and linear-quadratic-Gaussian/loop-transfer-recovery control synthesis, a distributed controller is designed from the linearized two-dimensional Navier-Stokes equations. This reduced-order controller, requiring only the wall-shear information, is shown to effectively suppress the linear disturbance in boundary layer flow under the uncertainty of Reynolds number. The controller also suppresses the nonlinear disturbance in the boundary layer flow, which would lead to unstable flow regime without control. The flow is relaminarized in the long run. Other effects of the controller on the flow are also discussed.

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Abbreviations

A,B,C,D:

Matrices associated with state-space equation

\(\hat A,\hat B,\hat C,D\) :

Matrices associated with reduced-order state-space equation

A c :

Loop coefficient matrix

A f :

Matrix associated with error evolution equation

A m :

Amplitude of disturbance

a mn,bmn,Cmn,dmn :

Coefficients of spectral decomposition

eα:

Error between internal state\(\hat x_\alpha \) and estimate state\(\tilde x_\alpha \)

F:

Matrix associated with cost criterion

g :

Forcing term in Navier-Stokes equation

h :

half channel height

I:

Identity matrix

\(\hat K_\alpha \) :

Control gain matrix

L :

4th order operator for linear term

L x :

Horizontal length

\(\hat L_\alpha \) :

Kaiman gain matrix

N :

4th order operator for nonlinear term

N a :

Matrix associated with cost criterion,\(\hat C_\alpha ^T \)

P :

Pressure

P α :

Conditional error variance

Q α :

Matrix associated with cost criterion,\(\hat C_\alpha ^T \hat C_\alpha \)

R α :

Matrix associated with cost criterion, Dα T αDα+FTαFα

Re:

Real part of complex number

Re :

Reynolds number based on half channel height

Re δ* :

Reynolds number based on displacement thickness

S α :

Solution of algebraic Riccati equation

t :

time

U :

Mean velocity in Poiseuille flow

U b :

Blasius mean velocity

U:

Input vector

U :

Disturbance velocity

\(\hat u(y)\) :

Eigenfunction of Orr-Sommerfeld solution

ũ:

Intermediate velocity

υ w :

Blowing/suction at wall

V, W:

Power spectral density of v and w

v, w:

White Gaussian noise associated with LQG

x :

State-space vector

\(\hat x\) :

Reduced-order state space vector

\(\tilde x\) :

Conditional mean estimate of\(\hat x\)

x, y :

Physical coordinate

y max :

Height of physical domain of boundary layer flow

Z t :

Measurement history

z:

Output vector

z :

Wall-shear measurement

α :

Wavenumber

αlβlγl,Sl :

Coefficients associated with RK3

β},ρ :

Tuning parameter for LQG/LTR

Γ α :

Input matrix associated with LQG problem

Γ m :

Combination of Chebyshev polynomial

Δ :

Laplacian operator

δ * :

Displacement thickness

δ i, :

Kronecker delta function

T:

Cost criterion

ψ :

Streamfunction

η max :

Height of computational domain of boundary layer flow

φ :

Pseudo pressure

σ :

Parameter to determine the grid concentration

υ :

Kinematic viscosity

σ α :

Filter residual

ω :

Eigenvalue of Orr-Sommerfeld solution

n :

Time step

i, j :

Index representing horizontal and wall-normal direction ( = 1, 2)

n :

Wavenumber index

l :

RK substep ( = 1, 2, 3)

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  1. Department of Mechanical & Aerospace Engineering, University of California, 90095, Los Angeles, California, USA

    Keun Hyoung Lee

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Lee, K.H. Control of boundary layer flow transition via distributed. KSME International Journal 16, 1561–1575 (2002). https://doi.org/10.1007/BF03021658

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  • DOI: https://doi.org/10.1007/BF03021658

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