Wavenumber–frequency spectrum model of fluctuating pressure on flat plate and circular cylinder in compressible flow

This study develops wavenumber–frequency spectrum models for wall-pressure fluctuations in turbulent boundary layers on flat plates and cylinders in compressible flow. Through non-dimensionalisation and solution of the momentum and continuity equations, a unified physical framework integrating near-field pressure and far-field acoustic regions is established, extending Lighthill’s acoustic analogy. Starting from the fluctuating pressure governing equation, Mach number effects in the acoustic region are explicitly introduced for the first time, and unified analytical expressions across the full wavenumber range are derived for both geometries. The proposed models achieve high-precision prediction across the entire wavenumber domain. Validation is performed via cylinder wind tunnel experiments at Mach 0.12 <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> <mml:mn>0.12</mml:mn> </mml:math> $0.12$ – 0.18 <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> <mml:mn>0.18</mml:mn> </mml:math> $0.18$ and flat-plate direct numerical simulation (DNS) at Mach 0.1 <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> <mml:mn>0.1</mml:mn> </mml:math> $0.1$ – 0.5 <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> <mml:mn>0.5</mml:mn> </mml:math> $0.5$ . Compared with classical models such as Chase II, Smol’yakov and Corcos, the present model shows better agreement with experimental and DNS data, particularly in low-wavenumber and acoustic regions, improving prediction accuracy by more than 10 <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> <mml:mn>10</mml:mn> </mml:math> $10$ dB. Key findings: (i) acoustic amplitude highly correlates with Mach number, while the convective ridge does not; (ii) the acoustic of flat-plate boundary is defined by k 1 squared plus k 3 squared minus left parenthesis k 0 minus upper M a k 1 right parenthesis squared equals 0 <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> <mml:msubsup> <mml:mi>k</mml:mi> <mml:mrow> <mml:mn>1</mml:mn> </mml:mrow> <mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:msubsup> <mml:mo>+</mml:mo> <mml:msubsup> <mml:mi>k</mml:mi> <mml:mrow> <mml:mn>3</mml:mn> </mml:mrow> <mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:msubsup> <mml:mo>−</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mi>k</mml:mi> <mml:mrow> <mml:mn>0</mml:mn> </mml:mrow> </mml:msub> <mml:mo>−</mml:mo> <mml:mi>M</mml:mi> <mml:mi>a</mml:mi> <mml:msub> <mml:mi>k</mml:mi> <mml:mrow> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> <mml:msup> <mml:mo stretchy="false">)</mml:mo> <mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:math> $k_{1}^{2} + k_{3}^{2} - (k_{0} - Mak_{1})^{2} = 0$ , where k <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="htt