{"id":592,"date":"2025-01-21T00:53:00","date_gmt":"2025-01-21T08:53:00","guid":{"rendered":"https:\/\/dornsife.usc.edu\/scribe\/?p=592"},"modified":"2025-10-21T00:56:00","modified_gmt":"2025-10-21T07:56:00","slug":"hydrodynamic-stability-three-dimensional-linear-stability-and-input-output-analysis-of-parallel-shear-flows","status":"publish","type":"post","link":"https:\/\/dornsife.usc.edu\/scribe\/2025\/01\/21\/hydrodynamic-stability-three-dimensional-linear-stability-and-input-output-analysis-of-parallel-shear-flows\/","title":{"rendered":"Hydrodynamic Stability: Three-Dimensional Linear Stability and Input-Output Analysis of Parallel Shear Flows"},"content":{"rendered":"\n\n\n\n\n \n \n\n\n\n\n\n\n\n
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Hydrodynamic Stability: Three-Dimensional Linear Stability and Input-Output Analysis of Parallel Shear Flows<\/h1>\n\n\n<\/div>\n \n \n By<\/span>Oliver Khan<\/strong>\n \n January 21, 2025<\/span>\n <\/div>\n<\/div>\n\n\n <\/div><\/div>\n\n \n \n\n\n\n\n\n\n\n
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Introduction:<\/strong><\/p>\n

Welcome back. Today, I am excited to return to our series on hydrodynamic stability. In previous blogs, we considered the stability of two-dimensional parallel shear flows. Now, we will derive the governing equations for stability analysis of three-dimensional parallel shear flows. Our mathematical work here will give us the tools to consider oblique modes and three-dimensional instabilities that are characteristic of channel flows. We will also have the opportunity to derive the Orr-Sommerfeld equation in physical space, rather than Fourier space, to get more practice manipulating the Navier-Stokes equations using vector calculus operations. This Orr-Sommerfeld equation will be coupled with the\u00a0Squire equation<\/em>\u00a0for the wall-normal vorticity, which will allow us to study three-dimensional effects. The resulting system of the Orr-Sommerfeld and Squire equations will form a dynamical system for the wall-normal velocity and vorticity, which will allow us to compute instability modes of the unforced flow, and also map input disturbances to output amplifications in velocity [1]. The resulting flow structures identified from this analysis can be beneficial in designing control strategies that encourage or suppress turbulence in channel flows.<\/p>\n

Orr-Sommerfeld Equation in Physical Coordinates:<\/strong><\/p>\n

Consider a base flow of the form\u00a0U<\/strong>(y) = U(y)\u00a0e<\/strong>x<\/sub>, where y is the wall-normal coordinate and\u00a0e<\/strong>x<\/sub>\u00a0is the unit vector in the streamwise direction. Any base flow of this form will be called a\u00a0parallel shear flow<\/em>, because the only velocity component is in the streamwise direction, and varies in the cross-stream direction. Now, consider infinitesimal disturbances of the form\u00a0u<\/strong>(x, y, z, t) = [u\u00a0e<\/strong>x<\/sub>\u00a0+ v\u00a0e<\/strong>y<\/sub>\u00a0+ w\u00a0e<\/strong>z<\/sub>](x, y, z, t), where x is the streamwise coordinate and z is the spanwise coordinate. Assume that all quantities are non-dimensionalized. If we assume very small disturbances, then the non-dimensionalized Navier-Stokes equations can be linearized around the base flow to yield the following:<\/p>\n

\u2202t<\/sub>u + U \u2202x<\/sub>u + U\u2019 v = -\u2202x\u00a0<\/sub>p + 1\/Re \u2206u + Fx<\/sub>\u00a0\u00a0 \u00a0 \u00a0 (x-momentum)<\/p>\n

\u2202t<\/sub>v + U \u2202x<\/sub>v = -\u2202y\u00a0<\/sub>p + 1\/Re \u2206v + Fy<\/sub>\u00a0\u00a0 \u00a0 \u00a0 (y-momentum)<\/p>\n

\u2202t<\/sub>w + U \u2202x<\/sub>w = -\u2202z\u00a0<\/sub>p + 1\/Re \u2206w + Fz<\/sub>\u00a0\u00a0 \u00a0 \u00a0 (z-momentum)<\/p>\n

\u2202x<\/sub>u + \u2202y<\/sub>v + \u2202z<\/sub>w = 0,\u00a0 \u00a0 \u00a0 \u00a0 (continuity)<\/p>\n

where U\u2019 = dU\/dy, Re is the Reynolds number, \u2206 = \u2202xx<\/sub>2<\/sup>\u00a0+ \u2202yy<\/sub>2<\/sup>\u00a0+ \u2202zz<\/sub>2<\/sup>\u00a0is the Laplacian, and Fx<\/sub>, Fy<\/sub>, and Fz<\/sub>\u00a0are input forcing terms in the x, y, and z directions, respectively. To reduce our problem from four variables (u, v, w, p) to three variables (u, v, w), we can eliminate pressure from the above equations. We do this by taking the divergence of the (vector) momentum equation, which results in the following equation for the Laplacian of pressure:<\/p>\n

-\u2206p + \u2202x<\/sub>Fx<\/sub>\u00a0+ \u2202y<\/sub>Fy<\/sub>\u00a0+ \u2202z<\/sub>Fz<\/sub>\u00a0= \u2202x<\/sub>\u00a0[U \u2202x<\/sub>u + U\u2019 v] + \u2202y<\/sub>\u00a0[U \u2202x<\/sub>v] + \u2202z<\/sub>\u00a0[U \u2202x<\/sub>w] =<\/p>\n

= 2U\u2019 \u2202x<\/sub>v + U \u2202x<\/sub>\u00a0[\u2202x<\/sub>u + \u2202y<\/sub>v + \u2202z<\/sub>w].<\/p>\n

Recognizing that \u2202x<\/sub>u + \u2202y<\/sub>v + \u2202z<\/sub>w = 0 yields the\u00a0pressure Poisson equation<\/em>:<\/p>\n

\u2206p = -2U\u2019 \u2202x<\/sub>v + \u2202x<\/sub>Fx<\/sub>\u00a0+ \u2202y<\/sub>Fy<\/sub>\u00a0+ \u2202z<\/sub>Fz<\/sub>.<\/p>\n

For computational efficiency, it is useful to cast the three equations for u, v, and w into just two equations for wall-normal velocity v and wall-normal vorticity \ud835\udf14y<\/sub>. To do this, let\u2019s first derive the Orr-Sommerfeld equation for v by taking the Laplacian of the y-momentum equation:<\/p>\n

\u2202t<\/sub>\u00a0\u2206v + \u2206 [U \u2202x<\/sub>v] = -\u2202y<\/sub>\u00a0\u2206p + 1\/Re \u22062<\/sup>v + \u2206Fy<\/sub>.<\/p>\n

The term \u2206 [U \u2202x<\/sub>v] can be simplified as follows:<\/p>\n

\u2206 (U \u2202x<\/sub>v) = U \u2202x<\/sub>( \u2202xx<\/sub>2<\/sup>\u00a0+ \u2202zz<\/sub>2<\/sup>)v + \u2202yy<\/sub>2<\/sup>\u00a0[U \u2202x<\/sub>v] =<\/p>\n

= U \u2202x<\/sub>( \u2202xx<\/sub>2<\/sup>\u00a0+ \u2202zz<\/sub>2<\/sup>)v + \u2202y<\/sub>\u00a0[U\u2019 \u2202x<\/sub>v + U \u2202xy<\/sub>2<\/sup>\u00a0v] =<\/p>\n

= U\u2019\u2019 \u2202x<\/sub>v + 2 U\u2019 \u2202xy<\/sub>2<\/sup>\u00a0v + U \u2202x<\/sub>\u00a0\u2206v.<\/p>\n

Therefore, the Laplacian of the y-momentum equation becomes:<\/p>\n

\u2202t<\/sub>\u00a0\u2206v + U\u2019\u2019 \u2202x<\/sub>v + 2 U\u2019 \u2202xy<\/sub>2<\/sup>\u00a0v + U \u2202x<\/sub>\u00a0\u2206v =<\/p>\n

-\u2202y<\/sub>\u00a0[-2U\u2019 \u2202x<\/sub>v + \u2202x<\/sub>Fx<\/sub>\u00a0+ \u2202y<\/sub>Fy<\/sub>\u00a0+ \u2202z<\/sub>Fz<\/sub>] + 1\/Re \u22062<\/sup>v + \u2206Fy<\/sub>\u00a0=<\/p>\n

2U\u2019\u2019 \u2202x<\/sub>v + 2U\u2019 \u2202xy<\/sub>2<\/sup>\u00a0v \u2013 \u2202xy<\/sub>2<\/sup>\u00a0Fx<\/sub>\u00a0\u2013 \u2202yy<\/sub>2<\/sup>\u00a0Fy<\/sub>\u00a0\u2013 \u2202yz<\/sub>2<\/sup>\u00a0Fz<\/sub>\u00a0+ 1\/Re \u22062<\/sup>v + \u2206Fy<\/sub>.<\/p>\n

With a little more simplification, we arrive at the Orr-Sommerfeld equation in physical space:<\/p>\n

\u2202t<\/sub>\u00a0\u2206v = -U \u2202x<\/sub>\u00a0\u2206v + U\u2019\u2019 \u2202x<\/sub>v + 1\/Re \u22062<\/sup>v \u2013 \u2202xy<\/sub>2<\/sup>\u00a0Fx<\/sub>\u00a0+ (\u2202xx<\/sub>2<\/sup>\u00a0+ \u2202zz<\/sub>2<\/sup>) Fy<\/sub>\u00a0\u2013 \u2202yz<\/sub>2<\/sup>\u00a0Fz<\/sub>.\u00a0 \u00a0 \u00a0 \u00a0 (Orr-Sommerfeld)<\/p>\n

Squire Equation for Wall-Normal Vorticity:\u00a0<\/strong><\/p>\n

The next step in formulating our stability problem is to derive an equation for the wall-normal vorticity, defined as:<\/p>\n

\ud835\udf14y<\/sub>\u00a0= \u2202z<\/sub>u \u2013 \u2202x<\/sub>w.<\/p>\n

To form our equation for \ud835\udf14y<\/sub>, we first take the z-derivative of the x-momentum equation:<\/p>\n

\u2202t<\/sub>\u00a0\u2202z\u00a0<\/sub>u + U \u2202x\u00a0<\/sub>\u2202z\u00a0<\/sub>u + U\u2019 \u2202z<\/sub>v = -\u2202x\u00a0<\/sub>\u2202z\u00a0<\/sub>p + 1\/Re \u2206 \u2202z<\/sub>u + \u2202z<\/sub>Fx<\/sub>.<\/p>\n

Now, take the x-derivative of the z-momentum equation:<\/p>\n

\u2202t<\/sub>\u00a0\u2202x\u00a0<\/sub>w + U \u2202x<\/sub>\u00a0\u2202x\u00a0<\/sub>w = -\u2202z\u00a0<\/sub>\u2202x\u00a0<\/sub>p + 1\/Re \u2206 \u2202x\u00a0<\/sub>w + \u2202x\u00a0<\/sub>Fz<\/sub>.<\/p>\n

Subtracting the x-derivative of the z-momentum equation from the z-derivative of the x-momentum equation yields:<\/p>\n

\u2202t<\/sub>\u00a0(\u2202z\u00a0<\/sub>u \u2013 \u2202x\u00a0<\/sub>w) + U \u2202x<\/sub>\u00a0(\u2202z\u00a0<\/sub>u \u2013 \u2202x\u00a0<\/sub>w) + U\u2019 \u2202z\u00a0<\/sub>v = 1\/Re \u2206 (\u2202z\u00a0<\/sub>u \u2013 \u2202x\u00a0<\/sub>w) + \u2202z\u00a0<\/sub>Fx<\/sub>\u00a0\u2013 \u2202x\u00a0<\/sub>Fz\u00a0<\/sub>,<\/p>\n

which simplifies to the Squire equation:<\/p>\n

\u2202t\u00a0<\/sub>\ud835\udf14y<\/sub>\u00a0= \u2013 U\u2019 \u2202z\u00a0<\/sub>v \u2013 U \u2202x\u00a0<\/sub>\ud835\udf14y<\/sub>\u00a0+ 1\/Re \u2206\ud835\udf14y<\/sub>\u00a0+ \u2202z\u00a0<\/sub>Fx<\/sub>\u00a0\u2013 \u2202x\u00a0<\/sub>Fz<\/sub>\u00a0,\u00a0 \u00a0 \u00a0 \u00a0 (Squire)<\/p>\n

Output Equations:<\/strong><\/p>\n

Using the Orr-Sommerfeld and Squire equations, we can compute instability modes and input-output relationships in terms of v and \ud835\udf14y<\/sub>. However, it is also useful to relate these quantities back to the streamwise and spanwise velocity components u and w. To form expressions for u and w in terms of v and \ud835\udf14y<\/sub>, we can compute the x and z-derivatives of \ud835\udf14y<\/sub>\u00a0and apply the continuity equation as follows:<\/p>\n

\u2202x<\/sub>\ud835\udf14y<\/sub>\u00a0=\u00a0 \u2202x<\/sub>\u00a0(\u2202z<\/sub>u \u2013 \u2202x<\/sub>w) = \u2202z<\/sub>\u00a0(- \u2202y<\/sub>v \u2013 \u2202z<\/sub>w) \u2013 \u2202xx<\/sub>2<\/sup>w = -\u2202zy<\/sub>2<\/sup>v \u2013 (\u2202xx<\/sub>2<\/sup>\u00a0+ \u2202zz<\/sub>2<\/sup>)w,<\/p>\n

\u2202z<\/sub>\ud835\udf14y<\/sub>\u00a0= \u2202z<\/sub>\u00a0(\u2202z<\/sub>u \u2013 \u2202x<\/sub>w) = \u2202zz<\/sub>2<\/sup>u \u2013 \u2202x<\/sub>\u00a0(-\u2202x<\/sub>u \u2013 \u2202y<\/sub>v) = \u2202xy<\/sub>2<\/sup>v + (\u2202xx<\/sub>2<\/sup>\u00a0+ \u2202zz<\/sub>2<\/sup>)u.<\/p>\n

Therefore, output equations for u and w are as follows:<\/p>\n

u = \u2013 (\u2202xx<\/sub>2<\/sup>\u00a0+ \u2202zz<\/sub>2<\/sup>)-1<\/sup>\u00a0(\u2202xy<\/sub>2<\/sup>v \u2013 \u2202z<\/sub>\ud835\udf14y<\/sub>),\u00a0 \u00a0 \u00a0 \u00a0 (u-output)<\/p>\n

w = \u2013 (\u2202xx<\/sub>2<\/sup>\u00a0+ \u2202zz<\/sub>2<\/sup>)-1<\/sup>\u00a0(\u2202yz<\/sub>2<\/sup>v + \u2202x<\/sub>\ud835\udf14y<\/sub>),\u00a0 \u00a0 \u00a0 \u00a0 (w-output)<\/p>\n

Linear Control System Formulation:<\/strong><\/p>\n

The Orr-Sommerfeld, Squire, and output equations can then be formally cast into the following state-space form, where F = [Fx<\/sub>\u00a0\u00a0 Fy<\/sub>\u00a0\u00a0 Fz<\/sub>]T<\/sup>\u00a0is the input vector, \ud835\udf13 = [v \u00a0 \ud835\udf14y<\/sub>]T<\/sup>\u00a0is the state vector, and \ud835\udf09 = [u \u00a0 v \u00a0 w]T<\/sup>\u00a0is the output vector:<\/p>\n

\u2202t<\/sub>\u00a0\ud835\udf13(x, y, z, t) = [\u0410 \ud835\udf13(t)](x, y, z) + [B F(t)](x, y, z)<\/p>\n

\ud835\udf09(x, y, z, t) = [C \ud835\udf13(t)](x, y, z).<\/p>\n

The operator A is defined as:<\/p>\n

\u0410 = [LOS<\/sub>, \u00a0 0;<\/p>\n

LC<\/sub>, \u00a0 LSQ<\/sub>],<\/p>\n

where the Orr-Sommerfeld, coupling, and Squire operators are respectively:<\/p>\n

LOS<\/sub>\u00a0= \u2206-1<\/sup>\u00a0(-U \u2202x<\/sub>\u00a0\u2206 + U\u2019\u2019 \u2202x<\/sub>\u00a0+ 1\/Re \u22062<\/sup>)<\/p>\n

LC<\/sub>\u00a0= -U\u2019 \u2202z<\/sub><\/p>\n

LSQ<\/sub>\u00a0= -U \u2202x<\/sub>\u00a0+ 1\/Re \u2206.<\/p>\n

The input matrix is:<\/p>\n

B = [- \u2206-1<\/sup>\u00a0\u2202xy<\/sub>2<\/sup>,\u00a0 \u2206-1<\/sup>\u00a0(\u2202xx<\/sub>2<\/sup>\u00a0+ \u2202zz<\/sub>2<\/sup>),\u00a0 \u00a0 \u2013 \u2206-1<\/sup>\u00a0\u2202yz<\/sub>2<\/sup>;<\/p>\n

\u2202z\u00a0<\/sub>, \u00a0 0, \u00a0 \u2013 \u2202x<\/sub>].<\/p>\n

Lastly, the output matrix is:<\/p>\n

C = [- (\u2202xx<\/sub>2<\/sup>\u00a0+ \u2202zz<\/sub>2<\/sup>)-1<\/sup>\u00a0\u2202xy<\/sub>2<\/sup>, \u00a0 (\u2202xx<\/sub>2<\/sup>\u00a0+ \u2202zz<\/sub>2<\/sup>)-1<\/sup>\u00a0\u2202z<\/sub>;<\/p>\n

1, \u00a0 0;<\/p>\n

\u2013 (\u2202xx<\/sub>2<\/sup>\u00a0+ \u2202zz<\/sub>2<\/sup>)-1<\/sup>\u00a0\u2202zy<\/sub>2<\/sup>, \u2013 (\u2202xx<\/sub>2<\/sup>\u00a0+ \u2202zz<\/sub>2<\/sup>)-1<\/sup>\u00a0\u2202x<\/sub>].<\/p>\n

A common set of boundary conditions for these equations, arising from the no-slip and continuity equations at the walls at y = \u00b1 1 are:<\/p>\n

v = \u2202y<\/sub>v = \ud835\udf14y<\/sub>\u00a0= 0 at y = \u00b1 1 for all x and z.<\/p>\n

If the flow is homogeneous in the x and z directions, then periodic boundary conditions can be used in x and z. This state-space model can serve several purposes. Since A is in block-diagonal form, the eigenvalues of A are determined by the eigenvalues of LOS<\/sub>\u00a0and LSQ<\/sub>. Therefore, the three-dimensional instability modes of channel flows can be separated between modes arising from the Orr-Sommerfeld equation and modes arising from the Squire equation. Furthermore, the impact of disturbances on the velocity field can be assessed using the transfer function,<\/p>\n

H(\ud835\udf14) = C (i\ud835\udf14 \u2013 A)-1<\/sup>\u00a0B.<\/p>\n

A singular value decomposition of the transfer function reveals that the most amplified disturbances come in the form of streamwise vortices and streaks, rather than the unstable eigenmodes. This is a reflection of the non-normality of the operator A, which arises both from the non-normality of LOS<\/sub>\u00a0and LSQ<\/sub>, and from the off-diagonal coupling term LC<\/sub>.<\/p>\n

Following [2], the transfer function H(\ud835\udf14) can be split into six different components, which map each input (Fx<\/sub>, Fy<\/sub>, Fz<\/sub>) to each velocity output (u,v,w):<\/p>\n

H(\ud835\udf14) = [Hux<\/sub>(\ud835\udf14), \u00a0 Huy<\/sub>(\ud835\udf14), \u00a0 Huz<\/sub>(\ud835\udf14);<\/p>\n

Hvx<\/sub>(\ud835\udf14), \u00a0 Hvy<\/sub>(\ud835\udf14), \u00a0 Hvz<\/sub>(\ud835\udf14);<\/p>\n

Hwx<\/sub>(\ud835\udf14), \u00a0 Hwy<\/sub>(\ud835\udf14), \u00a0 Hwz<\/sub>(\ud835\udf14)].<\/p>\n

Analysis of each of these components can provide significant insight into the precise directions of disturbances that amplify each of the velocity components, and therefore provide a more nuanced understanding than just analyzing H(\ud835\udf14) alone.<\/p>\n

Linear Control System in Fourier Space:<\/strong><\/p>\n

Finally, we mention that the Orr-Sommerfeld and Squire equations derived above can be Fourier transformed with respect to the homogeneous coordinates x and z. This yields a dynamical system in terms of the streamwise wavenumber kx<\/sub>\u00a0and spanwise wavenumber kz<\/sub>. In this case, the input is the Fourier-transformed forcing vector F(kx<\/sub>, kz<\/sub>) = [Fx<\/sub>\u00a0\u00a0 Fy<\/sub>\u00a0\u00a0 Fz<\/sub>]T<\/sup>(kx<\/sub>, kz<\/sub>), the state is the Fourier-transformed wall-normal velocity and vorticity \ud835\udf13(kx<\/sub>, kz<\/sub>) = [v \u00a0 \ud835\udf14y<\/sub>]T<\/sup>(kx<\/sub>, kz<\/sub>), and the output is the Fourier-transformed velocity field \ud835\udf09(kx<\/sub>, kz<\/sub>) = [u \u00a0 v \u00a0 w]T<\/sup>(kx<\/sub>, yz<\/sub>) is the output vector. The matrix equation is again:<\/p>\n

\u2202t<\/sub>\u00a0\ud835\udf13(kx<\/sub>, y, kz<\/sub>, t) = [\u0410(kx<\/sub>, kz<\/sub>) \ud835\udf13(kx<\/sub>, kz<\/sub>, t)](y) + [B(kx<\/sub>, kz<\/sub>) F(kx<\/sub>, kz<\/sub>, t)](y)<\/p>\n

\ud835\udf09(kx<\/sub>, y, kz<\/sub>, t) = [C(kx<\/sub>, kz<\/sub>) \ud835\udf13(kx<\/sub>, kz<\/sub>, t)](y).<\/p>\n

However, the operator A is defined as:<\/p>\n

\u0410(kx<\/sub>, kz<\/sub>) = [LOS<\/sub>(kx<\/sub>, kz<\/sub>), \u00a0 0;<\/p>\n

LC<\/sub>(kx<\/sub>, kz<\/sub>), \u00a0 LSQ<\/sub>(kx<\/sub>, kz<\/sub>)],<\/p>\n

where the Orr-Sommerfeld, coupling, and Squire operators are respectively:<\/p>\n

LOS<\/sub>(kx<\/sub>, kz<\/sub>) = \u2206-1<\/sup>\u00a0(-U ikx<\/sub>\u00a0\u2206 + U\u2019\u2019 ikx<\/sub>\u00a0+ 1\/Re \u22062<\/sup>)<\/p>\n

LC<\/sub>(kx<\/sub>, kz<\/sub>) = -U\u2019 ikz<\/sub><\/p>\n

LSQ<\/sub>(kx<\/sub>, kz<\/sub>) = -U ikx<\/sub>\u00a0+ 1\/Re \u2206,<\/p>\n

where the Laplacian is now \u2206 = \u2202yy<\/sub>2<\/sup>\u00a0\u2013 kx<\/sub>2<\/sup>\u00a0\u2013 kz<\/sub>2<\/sup>.<\/p>\n

The input matrix is:<\/p>\n

B(kx<\/sub>, kz<\/sub>) = [- \u2206-1<\/sup>\u00a0ikx<\/sub>\u2202y<\/sub>,\u00a0 \u2013 \u2206-1<\/sup>\u00a0(kx<\/sub>2<\/sup>\u00a0+ kz<\/sub>2<\/sup>),\u00a0 \u00a0 \u2013 \u2206-1<\/sup>\u00a0ikz<\/sub>\u2202y<\/sub>;<\/p>\n

ikz\u00a0<\/sub>, \u00a0 0, \u00a0 \u2013 ikx<\/sub>].<\/p>\n

Lastly, the output matrix is:<\/p>\n

C(kx<\/sub>, kz<\/sub>) = [(kx<\/sub>2<\/sup>\u00a0+ kz<\/sub>2<\/sup>)-1<\/sup>\u00a0ikx<\/sub>\u2202y<\/sub>, \u00a0 \u2013 (kx<\/sub>2<\/sup>\u00a0+ kz<\/sub>2<\/sup>)-1<\/sup>\u00a0ikz<\/sub>;<\/p>\n

1, \u00a0 0;<\/p>\n

\u2013 (kx<\/sub>2<\/sup>\u00a0+ kz<\/sub>2<\/sup>)-1<\/sup>\u00a0ikz<\/sub>\u2202y<\/sub>, (kx<\/sub>2<\/sup>\u00a0+ kz<\/sub>2<\/sup>)-1<\/sup>\u00a0ikx<\/sub>].<\/p>\n

In future blog posts, we will explore the discoveries made from these equations in more detail. Until then, please take care.<\/p>\n\n\n\n<\/div>\n\n\n <\/div><\/div>\n\n\n\n\n \n \n\n\n\n\n\n\n\n

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References:<\/strong><\/h6>\n
[1] M. R. Jovanovic and B. Bamieh.\u00a0The spatio-temporal impulse response of the linearized Navier-Stokes equations<\/a>. In\u00a0Proceedings of the 2001 American Control Conference<\/em>, Arlington, VA, pages 1948-1953, 2001.
\n[2] M. R. Jovanovic and B. Bamieh.\u00a0Componentwise energy amplification in channel flows<\/a>.\u00a0J. Fluid Mech.<\/em>, 534:145-183, July 2005.<\/h6>\n\n\n\n<\/div>\n\n\n <\/div><\/div>\n\n\n\n \n \n \n \n\n\n\n\n\n\n\n
\n\n \n
\n \n
\n\n \n

\n Read more from An Introduction to Flight:\n <\/h2>\n\n\n<\/div>\n \n
\n \n
\n\n \n

\n Hydrodynamic Stability: The Orr-Sommerfeld Equation<\/a>\n <\/h3>\n\n\n<\/div>\n <\/article>\n
\n \n
\n\n \n

\n Some Brief Words on Nonlinear Dynamical Systems and Chaos<\/a>\n <\/h3>\n\n\n<\/div>\n <\/article>\n
\n \n
\n\n \n

\n The Mathematics of Aerospace Engineering: Part 2 (Multivariable Calculus & Intro to Partial Differential Equations)<\/a>\n <\/h3>\n\n\n<\/div>\n <\/article>\n <\/div>\n\n\n <\/div><\/div>\n","protected":false},"excerpt":{"rendered":"","protected":false},"author":1018,"featured_media":311,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[34,33],"tags":[36,35,43],"class_list":["post-592","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-an-introduction-to-flight","category-past-editor-blogs","tag-an-introduction-to-flight","tag-oliver-khan","tag-spring-2025"],"acf":[],"yoast_head":"\nHydrodynamic Stability: Three-Dimensional Linear Stability and Input-Output Analysis of Parallel Shear Flows<\/title>\n<meta name=\"description\" content=\"An Introduction to Flight by Editor Oliver Khan\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/dornsife.usc.edu\/scribe\/2025\/01\/21\/hydrodynamic-stability-three-dimensional-linear-stability-and-input-output-analysis-of-parallel-shear-flows\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Hydrodynamic Stability: Three-Dimensional Linear Stability and Input-Output Analysis of Parallel Shear Flows - 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