{"id":582,"date":"2024-11-07T00:45:00","date_gmt":"2024-11-07T08:45:00","guid":{"rendered":"https:\/\/dornsife.usc.edu\/scribe\/?p=582"},"modified":"2025-10-21T00:46:48","modified_gmt":"2025-10-21T07:46:48","slug":"hydrodynamic-stability-resolvent-analysis","status":"publish","type":"post","link":"https:\/\/dornsife.usc.edu\/scribe\/2024\/11\/07\/hydrodynamic-stability-resolvent-analysis\/","title":{"rendered":"Hydrodynamic Stability: Resolvent Analysis"},"content":{"rendered":"\n\n\n\n\n \n \n\n\n\n\n\n\n\n
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Hydrodynamic Stability: Resolvent Analysis<\/h1>\n\n\n<\/div>\n \n \n By<\/span>Oliver Khan<\/strong>\n \n November 7, 2024<\/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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Hydrodynamic Stability: Resolvent Analysis<\/strong><\/p>\n

Welcome back. In today\u2019s blog post, we will continue our discussion of hydrodynamic stability by discussing resolvent analysis, which is complementary to classical linear stability analysis. While linear stability analysis looks at the growth of small perturbations to an unforced base flow, resolvent analysis finds maximum gain input-output structures in a flow field that describe the linear response of the flow to forcing. One of the revolutionary parts to resolvent analysis is that the forcing could be the nonlinear terms in the Navier-Stokes equations. Thus, this method can extract fundamental structures in turbulent flow fields by examining the response of the linear part of the Navier-Stokes equations to the nonlinear part. In this blog, we will begin by introducing the basic formulation of resolvent analysis and compare it to the formulation of classical linear stability analysis. We will then show the connection between high-gain resolvent modes and pseudospectra, and conclude with a brief overview of the applications of resolvent analysis.<\/p>\n

Resolvent analysis considers a base flow with velocity\u00a0U<\/strong>\u00a0and pressure P, and a disturbance (which is not necessarily small) with velocity\u00a0u<\/strong>\u00a0and pressure p. For the sake of generality, let\u2019s also assume an external forcing,\u00a0g<\/strong>. In this case, the nonlinear disturbance equations for the perturbation are<\/p>\n

\u2202u<\/strong>\/\u2202t\u00a0 = \u2013\u00a0U<\/strong>\u2022\u2207u<\/strong>\u00a0\u2013\u00a0u<\/strong>\u2022\u2207U<\/strong>\u00a0+ 1\/Re \u22072<\/sup>u\u00a0<\/strong>\u2013 \u2207p\u00a0 \u2013\u00a0u<\/strong>\u2022\u2207u<\/strong>\u00a0+\u00a0g<\/strong>,<\/p>\n

\u2207 \u2022\u00a0u<\/strong>\u00a0= \u2207T<\/sup>u<\/strong>\u00a0=<\/strong>\u00a00.<\/p>\n

If the disturbance velocity\u00a0u<\/strong>\u00a0and pressure p are combined into a column vector\u00a0q<\/strong>\u00a0= [u<\/strong>, p]T<\/sup>, then the momentum and continuity equations can be rewritten in block matrix form as<\/p>\n

\u2202\/\u2202t Eq<\/strong>\u00a0= Lq<\/strong>\u00a0+\u00a0f<\/strong>,<\/p>\n

with the singular matrix<\/p>\n

E = [I, 0;<\/p>\n

0, 0],<\/p>\n

the linearized Navier-Stokes operator<\/p>\n

L = [-U<\/strong>\u2022\u2207 \u2013 \u2207U<\/strong>T<\/sup>\u00a0+ 1\/Re \u22072<\/sup>, -\u2207;<\/p>\n

\u2207T<\/sup>, 0],<\/p>\n

and forcing vector<\/p>\n

f<\/strong>\u00a0= [-u<\/strong>\u2022\u2207u<\/strong>\u00a0+\u00a0g<\/strong>, 0]T<\/sup>.<\/p>\n

Since resolvent analysis typically works in the frequency domain, the next step is to assume harmonic forcing of the form\u00a0f<\/strong>\u00a0=\u00a0f<\/strong>\ud835\udf14<\/sub>\u00a0exp(i\ud835\udf14t). Since L and E are linear operators, this will produce a response\u00a0q<\/strong>\u00a0=\u00a0q<\/strong>\ud835\udf14<\/sub>\u00a0exp(i\ud835\udf14t), governed by the equation<\/p>\n

(i\ud835\udf14E \u2013 L)q<\/strong>\u00a0=\u00a0f<\/strong>.<\/p>\n

Therefore, the velocity and pressure field response\u00a0\u00a0q<\/strong>\u00a0can be written in terms of the forcing term\u00a0f<\/strong>\u00a0as<\/p>\n

q<\/strong>\u00a0= (i\ud835\udf14E \u2013 L)-1<\/sup>\u00a0f<\/strong>\u00a0:= H(i\ud835\udf14)\u00a0f<\/strong>.<\/p>\n

The map H(i\ud835\udf14) = (i\ud835\udf14E \u2013 L)-1<\/sup>\u00a0between the forcing and the response is important enough to be given its own name: the\u00a0resolvent<\/em>. Although H(i\ud835\udf14) is an infinite-dimensional operator that involves gradient and Laplacian operators, it can be discretized on a grid to yield a finite-dimensional representation. With a bit more work, we could then show that a singular value decomposition of the discretized resolvent operator,<\/p>\n

H(i\ud835\udf14) = U\ud835\udef4V*<\/sup>,<\/p>\n

gives the maximum-gain input-output modes in the flow field. In particular, the right singular vectors,\u00a0v<\/strong>, are the forcing modes, the left singular vectors,\u00a0u<\/strong>, are response modes, and the singular values, \ud835\udf0e, represent the amplification of\u00a0u<\/strong>\u00a0with respect to\u00a0v<\/strong>. Since the first couple singular values are often much larger than the others, the first couple forcing and response modes dominate the input-output dynamics. Therefore, the effect of the nonlinear term,\u00a0u<\/strong>\u2022\u2207u,<\/strong>\u00a0or the external forcing,\u00a0g,<\/strong>\u00a0on the flow field can often be understood by just analyzing the first couple forcing and response mode pairs. This opens the way for new understandings of turbulent flows via coherent structures or reduced-order models that can be used for control.<\/p>\n

At this point, two remarks are in order. First, the input-output dynamics described by the resolvent H(i\ud835\udf14) are separate but complementary to the internal dynamics described by linear stability analysis. While resolvent analysis considered the Navier-Stokes equations with forcing\u00a0f<\/strong>\u00a0comprising the internal convective nonlinearity and external forcing, the governing equation for linear stability analysis is<\/p>\n

\u2202\/\u2202t Eq<\/strong>\u00a0= Lq<\/strong>.<\/p>\n

When\u00a0q<\/strong>\u00a0is assumed to have the form\u00a0q<\/strong>\ud835\udf06<\/sub>\u00a0exp(\ud835\udf06t), the equation above yields the generalized eigenvalue problem<\/p>\n

\ud835\udf06 Eq<\/strong>\ud835\udf06<\/sub>\u00a0= Lq<\/strong>\ud835\udf06<\/sub>,<\/p>\n

which yields the instability modes\u00a0q<\/strong>\ud835\udf06<\/sub>, and their frequencies Im(\ud835\udf06) and growth rates Re(\ud835\udf06). The second remark is that the resolvent operator H(i\ud835\udf14) is closely related to the pseudospectrum of L. In particular, we can recall in the last blog post that the \ud835\udf00-pseudospectrum of L is the set of all complex numbers z such that ||(zE \u2013 L)-1<\/sup>|| \u2265 \ud835\udf00-1<\/sup>. Therefore, the pseudospectrum is equivalently defined as the set of complex numbers z such that ||H(z)||\u00a0 \u2265 \ud835\udf00-1<\/sup>, which can itself be rewritten as the set of complex numbers z that satisfy \ud835\udf0emax<\/sub>(H(z)) \u2265 \ud835\udf00-1<\/sup>, where \ud835\udf0emax<\/sub>\u00a0is the maximum singular value of H(z). Therefore, in resolvent analysis, the operator H is evaluated along the imaginary axis, and the maximum singular value of H(i\ud835\udf14) is directly related to the pseudospectrum where \ud835\udf00 = 1\/\ud835\udf0emax<\/sub>. If H(i\ud835\udf14) has large values of \ud835\udf0emax<\/sub>\u00a0across a range of frequencies that lie far away from the eigenvalues of L, that means that the pseudospectra of L extend far away from the eigenvalues of L. The only way this can happen is if L is non-normal, and supports transient energy growth. Hence, the high-gain input-output mode pairs identified from resolvent analysis can have high gain due to non-normality of L.<\/p>\n

Although resolvent analysis has a lot of rich and dense mathematics, it has proven to be a valuable tool in a multitude of settings. When external forcing is neglected and\u00a0f<\/strong>\u00a0is taken to be just the nonlinear terms from the Navier-Stokes equation, then the structures of the forcing and response modes can provide insight into fundamental processes in turbulence. If the nonlinear terms are neglected but the external forcing is taken to be nonzero, then the forcing modes give the forcing structures that produce the most amplified responses, which are given by the response modes. This information can then be used to design actuators that excite a flow field in the locations marked by the high-gain forcing modes. Next week, we will explore more applications of resolvent analysis and provide a bit more details on how reduced-order models can be constructed in the resolvent framework. 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\n

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\n Read more from An Introduction to Flight:\n <\/h2>\n\n\n<\/div>\n \n
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\n Hydrodynamic Stability: Background and Basic Definitions<\/a>\n <\/h3>\n\n\n<\/div>\n <\/article>\n
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\n Hydrodynamic Stability: General Form of the Linearized Disturbance Equations<\/a>\n <\/h3>\n\n\n<\/div>\n <\/article>\n
\n \n
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\n Hydrodynamic Stability: Non-Normality and Transient Energy Growth<\/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":[36,42,35],"class_list":["post-582","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-fall-2024","tag-oliver-khan"],"acf":[],"yoast_head":"\nHydrodynamic Stability: Resolvent Analysis<\/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\/2024\/11\/07\/hydrodynamic-stability-resolvent-analysis\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Hydrodynamic Stability: Resolvent Analysis - 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