Welcome back. In this post, we will continue our discussion of hydrodynamic stability by exploring the linearized Navier-Stokes equations in three dimensions. These equations are important because they describe how tiny disturbances grow in a fluid flow and ultimately lead to turbulence. They are also the basis for more specialized stability equations applied in aerospace engineering. In particular, we will develop the tools to derive the Orr-Sommerfeld equation, which determines the frequencies and growth rates of disturbances that cause turbulence in boundary layers. Our derivations will also allow us to explore infinite-dimensional operators since the linearized Navier-Stokes equations can be cast as a dynamical system governed by an infinite-dimensional analog to a matrix. This concept is important because infinite-dimensional operators can be decomposed into eigenvalues and eigenfunctions just as finite-dimensional matrices can be decomposed into eigenvalues and eigenvectors. These eigen-decompositions are insightful because they provide the growth rates of the most unstable disturbances and their spatial structures in a flow. If unstable structures are concentrated in specific locations, then flow controllers can act on those locations to encourage or prevent the transition to turbulence.
One of the first steps to studying the growth of disturbances about a base flow with velocity, big U, and pressure, big P, is to consider the linear growth of tiny fluctuations with velocity, little u, and pressure, little p. Although the evolution of disturbances is most accurately described by the full nonlinear Navier-Stokes equations, perturbations that are small enough can be approximated sufficiently by the linearized Navier-Stokes equation. To derive the linearized equations that govern very tiny disturbances, we write the non-dimensionalized Navier-Stokes equations for the entire flow, with velocity U + u and pressure P + p as
∂/∂t (U + u) + (U + u) • ∇(U + u) = -∇(P + p) + 1/Re ∇2(U + u),
∇ • (U + u) = 0.
If we assume that the base flow U is a steady solution of the Navier-Stokes equation, then ∂U/∂t = 0 and
U • ∇U = -∇P + 1/Re ∇2U,
∇ • U = 0.
Now, let’s subtract these base flow equations from the full flow equations and expand the convective term (U + u) • ∇(U + u). This process yields
∂u/∂t + U • ∇u + u • ∇U + u • ∇u = -∇p + 1/Re ∇2u,
∇ • u = 0.
The second equation shows that the disturbances also must satisfy the continuity condition. While the first equation is nonlinear in terms of the fluctuation u, we can neglect the nonlinear term u • ∇u provided that u is very small. In this case, we arrive at the linearized 3D Navier-Stokes equations,
∂u/∂t = – U • ∇u – u • ∇U – ∇p + 1/Re ∇2u,
∇ • u = 0.
A few comments are in order here. First, the pressure term can be eliminated by projecting u onto a divergence-free basis. Therefore, the equations above can be written compactly as:
∂u/∂t = – U • ∇u – u • ∇U + 1/Re ∇2u,
as long as it is understood that u must be divergence-free. Further, the momentum equation has been written suggestively with the time derivative for u on the left side and all of the spatial gradients on the right side. This equation has the form
∂u/∂t = Lu,
which is an infinite-dimensional linear dynamical system that is governed by the operator:
L(·) = – U • ∇(·) – (·) • ∇U + 1/Re ∇2(·),
that is parameterized by the base flow U and Reynolds number Re. This formulation is important because it allows us to apply the powerful theory of linear dynamical systems to study the behavior of perturbations. Specifically, the eigenvalues ?j of L indicate the linear asymptotic growth or decay of disturbances, and the eigenfunctions ûj of L represent the flow structures that grow or decay at the rates given by the real parts of ?j. Next week, we will apply these concepts to study the Orr-Sommerfeld equation, which is derived from the linearized Navier-Stokes equations to describe 2D flows with one inhomogeneous direction. Until then, take care.
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