# The Relationship Between Surface Pressure Spectra and Vorticity in a
Turbulent Boundary Layer

Stewart Glegg1  Siddhartha Verma1,2  and Lyubov Denissova1<EMAIL_ADDRESS>1
Department of Ocean and Mechanical Engineering, Florida Atlantic University,
Boca Raton, FL 33431, USA 2 Harbor Branch Oceanographic Institute, Florida
Atlantic University, Fort Pierce, FL 34946, USA

###### Abstract

The modeling of surface pressure wave number spectra beneath a turbulent
boundary layer is reviewed and reconsidered in terms of the vorticity in the
flow. Using a solution based on the vorticity equation and Squires theorem,
which was originally given by Chase(1991), it is shown that the complete
solution for surface pressure spectrum can be specified using the non-linear
turbulence-turbulence interaction terms as sources. It is then shown that the
surface pressure can be directly related to the vorticity in the flow. The
results are checked against a Direct Numerical Simulation of a channel flow
configuration. It is shown that the vorticity associated with the wall shear
stress decays rapidly over a distance of about seven wall units and so the
outer vorticity dominates in the majority of the flow. The vorticity
correlation functions are evaluated and it is shown that the length scale of
the vorticity normal the wall is only about ten wall units so the flow can be
modeled as the superposition of uncorrelated vortex sheets. Finally a model
for the surface pressure wave number spectrum is developed by modeling the
vorticity spectrum based on the distribution of root mean square (rms)
vorticity in the flow, and an integral length scale that is about ten wall
units. Using inputs from the numerical simulation, a good fit to existing
empirical models for surface pressure spectra is obtained.

##  1 Introduction

The surface pressure fluctuations beneath a fully developed turbulent boundary
layer are important in many engineering applications. Surface pressure
perturbations can be the source of panel vibration on aircraft and marine
vehicles (Graham, 1997; Blake, 2017) and, using Amiet’s theory (Amiet, 1976),
the cause of trailing edge noise which is one of the most important noise
sources on aircraft airframes, fans, wind turbines and helicopter rotors. For
these reasons there has been considerable attention given to the prediction of
surface pressure spectra and wavenumber spectra, and how they are related to
the characteristics of the turbulent boundary layer (see Blake (2017) for a
recent review).

The approach to modeling the surface pressure spectrum below a turbulent
boundary layer has been based on the reduction of experimental measurements to
obtain empirical formulas (Graham, 1997; Chase, 1991; Goody, 2004), the use of
Direct Numerical Simulations (DNS) of a channel flow (Chang et al., 1999; Kim,
1989; Anantharamu & Mahesh, 2020), Large Eddy Simulations (LES) over rough
walls (Ji & Wang, 2012), or analytical modeling based on the solution to a
Poisson’s equation (Kraichnan, 1956) or Lighthill’s Acoustic Analogy (Ffowcs-
Williams, 1965). The modeling based on empirical formulas is limited to the
types of flow in which the measurements were made, which are almost
exclusively zero pressure gradient turbulent flows over smooth or rough walls.
Similarly the DNS and LES calculations have been limited to simple flows, and
are challenging in terms the computational storage requirements to ensure the
convergence of time and space averages. None of these approaches are able to
identify the flow characteristics that are responsible for the pressure
perturbations in a less restricted flow environment, such as the fully three
dimensional flow near the trailing edge or tip of a rotating fan blade.

To extend the empirical and computational models to a more complex flow type,
analytical models of the flow characteristics that cause the surface pressure
fluctuations are required. In general, most of the modeling to date has been
based on the use of Lighthill’s Acoustic Analogy, or a Poisson’s equation that
is the same as Lighthill’s equation in the incompressible limit. These models
have led to the development of the Blake-TNO model (Blake, 2017; Grasso et
al., 2019) which specify the surface pressure based on two source terms that
are derived from Lighthill’s stress tensor, $T_{ij}$. The first of these terms
is the so called fast term that depends on the mean velocity gradient normal
to the wall and the unsteady velocity component normal to the wall. This term
is linear in perturbation quantities, and most readily modeled using
homogeneous turbulence wavenumber spectrum models and an assumed uniform
convection speed (Blake, 2017). The second source term is based on the
turbulence-turbulence interactions which cause a non-linear quantity,
sometimes referred to as the slow term. The relative magnitude of these two
terms is hard to assess because the non-linear interactions are almost
impossible to measure or model, and can only be realistically obtained from
numerical approaches such as LES or DNS. Many studies (Grasso et al., 2019)
have argued that the linear term dominates in certain frequency ranges and
have modeled the surface pressure spectra based on the linear term alone. This
suggests that the surface pressure spectra are determined by the mean flow
gradient and the vertical component of the unsteady velocity fluctuations.
Then by using a suitable model for the wavenumber spectra of the turbulence in
the boundary layer, combined with a number of assumptions of the turbulence
lengthscales, and the rms turbulence velocity distribution obtained from RANS
calculations, a surface pressure spectrum model is obtained that is sensitive
to modifications in the mean flow.

The problem with this approach is that it assumes that the turbulent velocity
fluctuations are known precisely throughout the flow. If there is a DNS
calculation of the flow then the source terms for the Poisson’s equation can
be evaluated exactly and the distribution of ‘apparent’ or modelled source
strength defined. This was recently the focus of the study by Anantharamu &
Mahesh (2020), who used a DNS of a channel flow to define the terms on the
right side of the Poisson’s equation, and a Greens function to obtain the
apparent distribution of sources that caused the surface pressure
fluctuations. We use the term ‘apparent’ because, as was shown by Hariri &
Akylas (1985), the pressure fluctuations throughout the flow and the local
normal velocity are coupled, so one cannot be treated as a source for the
other. In their study of wall shear stress, Hariri & Akylas (1985) solved both
the Poisson’s equation for the pressure and the vertical component of the
momentum equation as two coupled partial differential equation with two
unknowns. The right side of these two equations were non-linear, turbulence-
turbulence interaction terms. It is often argued that this is a complete
solution of the Navier Stokes equations because, in the wavenumber frequency
domain, the non-linear terms are uncoupled with the linear terms and so may be
calculated separately as ‘true’ source terms.

Separating the linear terms in the Navier Stokes equations from the non-linear
terms for a parallel boundary layer flow to obtain the surface pressure was
first carried out by Landahl (1967) using the inhomogeneous form of the Orr-
Sommerfeld equation to solve for the vertical velocity fluctuations in terms
of non-linear source terms. The surface pressure was then obtained by
integrating the wall normal component of the momentum equation. A modal
solution to the Orr-Sommerfeld equations was used and the surface pressure
obtained as a modal expansion. All the modes were found to decay with
downstream distance and, for a homogeneous flow, the dominant modes were those
with phase speeds with the smallest imaginary parts. A similar approach was
used by Chase (1991) who re-cast the vorticity equation using Squires
transformation to obtain two uncoupled equations for the unsteady velocity in
a parallel shear flow. The non-linear terms were defined as source terms in
the frequency wavenumber domain, and the surface pressure and wall shear
stress specified directly from the momentum equations evaluated at the wall.
The source terms for the pressure fluctuations was then specified in terms of
only three components of the fluctuating Reynolds shear stress. Chase (1991)
solved the Orr-Sommerfeld equation in the limit that the streamwise wavenumber
tended to zero, and defined the vertical velocity perturbations and the
surface pressure in terms of the non-linear source terms by using the method
of variation of parameters.

In all these approaches the most difficult part of the modeling is to
correctly specify the non-linear source terms. Without recourse to LES or DNS
there are no empirical or semi-empirical models for these terms, or well
accepted insights as how they are a function of the mean flow and its
statistics. In this paper we will address this problem by simplifying Chase’s
(1991) analysis so that the source term is limited to the unsteady ‘outer’
vorticity in the flow. We will define this as the vorticity in the flow with
the vorticity caused by the wall shear stress subtracted. We will show
analytically, and by using DNS calculations, that the surface pressure
spectrum can be specified as an integral of the outer vorticity with an
exponentially decaying weighting factor. This has some important implications
for modeling more complex flows such as boundary layers with non-zero pressure
gradients, rough wall boundary layers and boundary layers downstream of
obstructions. In those flows modeling the convected vorticity is a tractable
problem. For many practical problems it may also have a local solution when
the scale of the vorticity is sufficiently small that it can be embedded into
a mean flow that changes on a much larger scale.

The layout of this paper is as follows. In section 2 we will discuss the
approach to surface pressure based on Lighthill’s Analogy and the Poisson’s
equation, specifically highlighting the role of viscous shear stress at the
wall. In section 3 we will specify the Poisson’s equation for a parallel shear
flow in the wavenumber frequency domain and show how there are two coupled
equations that must be solved for both the fluctuating pressure and the
vertical velocity component. In section 4 we will use the vorticity equation
and Squires transformation to derive the inhomogeneous Orr-Sommerfeld equation
as was done by Chase (1991), and relate the surface pressure to the viscous
shear stress at the wall. We then specify the inner and outer vorticity and
show how the surface pressure can be specified in terms of the outer vorticity
alone. In section 5 we use a DNS calculation for a channel flow to verify the
analytical results given in section 4. In section 6 we specify the wavenumber
spectrum of the surface pressure based on simple model of the vorticity
fluctuations in a turbulent boundary layer, and compare the predicted spectrum
to the Goody model for surface pressure spectra. In section 7 details are
given of the vorticity correlation functions obtained from the DNS data, and
it is shown that they are consistent with the simple model of the vorticty
spectrum used in section 6.

##  2 The Sources of Surface Pressure Fluctuations based on Poisson’s Equation

It is natural (Ffowcs-Williams, 1965) to start a derivation of the pressure
fluctuations in a flow by considering Lighthill’s Acoustic Analogy which
describes the propagation of acoustic waves in a stationary medium that are
generated by a region of turbulence. In a stationary medium the density
perturbations are given by the inhomogeneous wave equation:

$\dfrac{\partial^{2}\rho^{\prime}}{\partial
t^{2}}-c_{\infty}^{2}\dfrac{\partial^{2}\rho^{\prime}}{\partial
x_{i}^{2}}=\dfrac{\partial^{2}T_{ij}}{\partial x_{i}\partial x_{j}}$ (1)

and provided that the density is only a function of pressure and the speed of
sound is constant, as is the case in most flows of interest at low Mach
number, the acoustic variable on the right side of this equation is related to
the pressure perturbation $p$ as $p=\rho^{\prime}c_{\infty}^{2}$. However
Lighthill’s equation only gives the propagation of sound outside the turbulent
flow and so, in principle, we cannot use it in the flow to obtain the surface
pressure because $T_{ij}$ includes terms that depend on $\rho^{\prime}$ (Glegg
& Devenport, 2017). Nevertheless this approach has been used to calculate
surface pressure fluctuations beneath a turbulent boundary layer (Grasso et
al., 2019; Blake, 2017) and so we will review the background to this approach
before introducing alternatives.

In an incompressible flow Lighthill’s equation can be used to relate the
pressure perturbation to the stress tensor $T_{ij}$ by letting $c_{\infty}$
tend to infinity, which effectively eliminates the term that depends on the
time derivative and yields the Poisson’s equation

$\dfrac{\partial^{2}p}{\partial
x_{i}^{2}}=-\dfrac{\partial^{2}T_{ij}}{\partial x_{i}\partial x_{j}}$ (2)

The right side of this equation depends on both a turbulent shear stress $\rho
v_{i}v_{j}$ and viscous stress $\sigma_{ij}$ and can be expanded out as

$\dfrac{\partial^{2}T_{ij}}{\partial x_{i}\partial
x_{j}}=\dfrac{\partial^{2}\left(\rho v_{i}v_{j}\right)}{\partial x_{i}\partial
x_{j}}-\dfrac{\partial^{2}\sigma_{ij}}{\partial x_{i}\partial x_{j}}$ (3)

However, using the definition of viscous shear stress for an incompressible
flow ($\sigma_{ij}=\mu(v_{i,j}+v_{j,i})$, and $v_{i,i}=0$) we see that

$\dfrac{\partial^{2}\sigma_{ij}}{\partial x_{i}\partial
x_{j}}=\mu\dfrac{\partial^{3}v_{i}}{\partial x_{i}\partial x_{j}^{2}}=0$ (4)

so the viscous shear stress does not contribute to the source term in an
incompressible flow. The only remaining quantity depends on $\rho v_{i}v_{j}$,
where, because the flow is incompressible $\rho=\rho_{0}$ is constant. It is
also possible to write this part of the source term as the product of two
derivatives, so

$\dfrac{\partial^{2}T_{ij}}{\partial x_{i}\partial
x_{j}}=\dfrac{\partial^{2}\left(\rho_{0}v_{i}v_{j}\right)}{\partial
x_{i}\partial x_{j}}=\rho_{0}\dfrac{\partial}{\partial
x_{i}}\left(v_{i}\dfrac{\partial v_{j}}{\partial x_{j}}+v_{j}\dfrac{\partial
v_{i}}{\partial x_{j}}\right)=\rho_{0}\dfrac{\partial v_{j}}{\partial
x_{i}}\dfrac{\partial v_{i}}{\partial x_{j}}$ (5)

We can now separate the mean flow effects from the turbulent velocity
fluctuations by specifying $v_{i}=U_{i}+u_{i}$, where $U_{i}$ represents the
time invariant mean flow and $u_{i}$ the time varying part of the flow. For a
two dimensional parallel mean flow in the $x_{1}$ direction $U_{1}=U(x_{2})$
is only a function of $x_{2}$. In this case the source term becomes

$\dfrac{\partial^{2}T_{ij}}{\partial x_{i}\partial
x_{j}}=2\rho_{0}\dfrac{\partial U}{\partial x_{2}}\dfrac{\partial
u_{2}}{\partial x_{1}}+\rho_{0}\dfrac{\partial u_{j}}{\partial
x_{i}}\dfrac{\partial u_{i}}{\partial x_{j}}$ (6)

Note here that the factor of $2$ arises because of the repeated indices. The
two parts of the source term are sometimes referred to as the mean shear-
turbulence and the turbulence-turbulence interaction terms.

The important conclusion from these derivations is that the unsteady pressure
in an incompressible shear flow apparently depends on two different
mechanisms. The first is linear in fluctuating quantities and also depends on
the mean shear. It implies that if the flow is uniform so $\partial U/\partial
x_{2}=0$, then there are no pressure perturbations generated by linear
fluctuations, and only the non-linear terms are important. It also implies
that regions of high mean shear will result in larger pressure fluctuations.
The problem with this interpretation is that the mean shear and the turbulence
are not necessarily uncoupled, and knowledge of both is needed in order to
correctly identify the regions of the flow that cause the unsteady pressure.
The non-linear term is more complicated, as it depends on the interaction of
the turbulence with itself. In other words the turbulent eddies are convected
by not only the mean flow but also the turbulence that randomly changes the
direction of motion of the eddy. The random trajectory of the eddy results in
momentum flux that must be matched by a pressure perturbation, and this is
reflected in the non-linear source term. In a turbulent boundary layer both
the mean shear and the turbulent shear stress are largest close to the
surface, and so knowledge of both terms is needed to determine the surface
pressure. Some studies suggest that the mean shear term dominates (Blake,
2017), but others suggest that both terms are of the same magnitude, and
identifying their relative importance remains a challenge.

##  3 The Surface Pressure Wavenumber Spectrum

###  3.1 The Solution to the Poisson’s Equation for the Pressure

The applications that we are concerned with require the input to be specified
as the wavenumber spectrum of the surface pressure fluctuations (see Blake
(2017) and Grasso et al. (2019)). This is the quantity required to calculate
the noise from a rough surface or a trailing edge (Glegg & Devenport, 2017),
and similarly, the vibration of a panel caused by an attached turbulent
boundary layer (Blake, 2017). To obtain the surface pressure wavenumber
spectrum we apply Fourier transforms to obtain the pressure wavenumber
spectrum at a location $x_{2}$ as

${\accentset{\approx}{p}}(x_{2},k_{1},k_{3},\omega)=\dfrac{1}{\left(2\pi\right)^{3}}\int\displaylimits_{-T}^{T}\int\displaylimits_{-R}^{R}\int\displaylimits_{-R}^{R}p(\bm{x},t)e^{i\omega
t-ik_{1}x_{1}-ik_{3}x_{3}}dtdx_{1}dx_{3}$ (7)

Note that the pressure is now defined as a function of frequency $\omega$, the
wavenumbers $k_{1}$ and $k_{3}$, and the distance from the surface $x_{2}$.
The constants $R$ and $T$ both tend to infinity.

When this transform is applied to the Poisson’s equation for the pressure
perturbations in the flow (using equations 2 and 6), it becomes

$\dfrac{\partial^{2}\accentset{\approx}{p}}{\partial
x_{2}^{2}}-k_{s}^{2}\accentset{\approx}{p}=\accentset{\approx}{q}_{MS}(x_{2},k_{1},k_{3},\omega)+\accentset{\approx}{q}_{TT}(x_{2},k_{1},k_{3},\omega)$
(8)

Where $k_{s}=(k_{1}^{2}+k_{3}^{2})^{1/2}$ and the source terms are defined by

$\displaystyle\accentset{\approx}{q}_{MS}(x_{2},k_{1},k_{3},\omega)$
$\displaystyle=-\dfrac{1}{\left(2\pi\right)^{3}}\int\displaylimits_{-T}^{T}\int\displaylimits_{-R}^{R}\int\displaylimits_{-R}^{R}\left(2\rho_{0}\dfrac{\partial
U}{\partial x_{2}}\dfrac{\partial u_{2}}{\partial x_{1}}\right)e^{i\omega
t-ik_{1}x_{1}-ik_{3}x_{3}}dtdx_{1}dx_{3}$ (9a)
$\displaystyle\accentset{\approx}{q}_{TT}(x_{2},k_{1},k_{3},\omega)$
$\displaystyle=-\dfrac{1}{\left(2\pi\right)^{3}}\int\displaylimits_{-T}^{T}\int\displaylimits_{-R}^{R}\int\displaylimits_{-R}^{R}\left(\rho_{0}\dfrac{\partial
u_{j}}{\partial x_{i}}\dfrac{\partial u_{i}}{\partial x_{j}}\right)e^{i\omega
t-ik_{1}x_{1}-ik_{3}x_{3}}dtdx_{1}dx_{3}$ (9b)

The problem is now cast in the form of a second order inhomogeneous
differential equation (equation 8) that can most conveniently be solved by
using a Green’s function that is the solution to

$\dfrac{\partial^{2}g(x_{2},y)}{\partial
x_{2}^{2}}-k_{s}^{2}g(x_{2},y)=\delta(x_{2}-y)$ (10)

It is relatively easy to show that

$g(x_{2},y)=-\left(\dfrac{e^{-k_{s}\lvert
x_{2}-y\rvert}+Ae^{-k_{s}(x_{2}+y)}}{2k_{s}}\right)$ (11)

where $A$ is a constant defined by the boundary conditions. The pressure is
then given as

$\accentset{\approx}{p}(x_{2})=\int\displaylimits_{0_{+}}^{\infty}\left(\accentset{\approx}{q}_{MS}(y)+\accentset{\approx}{q}_{TT}(y)\right)g(x_{2},y)dy$
(12)

The surface pressure is readily obtained by setting $y>x_{2}=0$, so

$\accentset{\approx}{p}(0)=\int\displaylimits_{0_{+}}^{\infty}\dfrac{(1+A)e^{-k_{s}y}}{2k_{s}}\left(\accentset{\approx}{q}_{MS}(y)+\accentset{\approx}{q}_{TT}(y)\right)dy$
(13)

To determine the constant $A$ we consider the $x_{2}$ component of the
momentum equation on the surface, and apply the no slip boundary condition so
all the flow velocities are zero on the wall so

$\left[\dfrac{\partial\accentset{\approx}{p}}{\partial
x_{2}}\right]_{x_{2}=0}=\left[\mu\dfrac{\partial^{2}\accentset{\approx}{u}_{2}}{\partial
x_{2}^{2}}\right]_{x_{2}=0}$ (14)

The surface pressure gradient therefore depends on the second derivative of
the unsteady velocity normal to the wall. Without more detailed modeling this
is hard to specify, but the scaling of the flow at very high Reynolds numbers
would suggest that the viscous terms can be ignored, so the pressure gradient
is zero at the wall. In that case the derivative of the Green’s function
$g(x_{2},y)$ with respect to $x_{2}$ must be zero at $x_{2}=0$, when $y>0$,
and so $A=1$.

###  3.2 The Surface Pressure

The analysis given in the previous section assumed that the terms on the right
side of equation 8 were source terms, which implies that they are not related
to the pressure perturbation that is the dependent variable of the
differential operator on the left side of the equation. This is not the case
because the pressure and normal velocity are related by the momentum equation.
To show this consider the source terms given by equation 9 in more detail. The
mean shear part may be written as

$\accentset{\approx}{q}_{MS}(x_{2},k_{1},k_{3},\omega)=-2ik_{1}\rho_{0}U^{\prime}\accentset{\approx}{u}_{2}(x_{2},k_{1},k_{3},\omega)$
(15)

where $U^{\prime}$ is the mean shear $\partial U/\partial x_{2}$, and a prime
will be used to denote a differentiation with respect to $x_{2}$. This
notation will be used in the rest of the paper, but does not apply to
$\rho^{\prime}$ that only appears in §2.

Equation 15 shows that in the wavenumber frequency domain the mean shear
turbulence interaction term is proportional to $\accentset{\approx}{u}_{2}$,
which can also be related to the pressure perturbations by the component of
the momentum equation in the $x_{2}$ direction, given as

$\rho_{0}\dfrac{\partial u_{2}}{\partial t}+\rho_{0}U\dfrac{\partial
u_{2}}{\partial x_{1}}+\dfrac{\partial p}{\partial
x_{2}}-\mu\nabla^{2}u_{2}=-\rho_{0}u_{j}\dfrac{\partial u_{2}}{\partial
x_{j}}$ (16)

which, after applying the Fourier transforms defined above gives

$\accentset{\approx}{p}\
^{\prime}+ik_{1}\rho_{0}(U-c)\accentset{\approx}{u}_{2}-\mu\left(\accentset{\approx}{u}_{2}^{\prime\prime}-k_{s}^{2}\accentset{\approx}{u}_{2}\right)=\dfrac{\accentset{\approx}{q}_{2}}{ik_{1}}$
(17)

where $c=\omega/k_{1}$ is the phase speed, and $q_{2}$ is the non-linear
source term.

The Poisson’s equation for the pressure (equation 8) is then re-written by
substituting equation 15 and moving the mean shear part of the source term to
the left side so

$\accentset{\approx}{p}\
^{\prime\prime}-k_{s}^{2}\accentset{\approx}{p}+2ik_{1}\rho_{0}U^{\prime}\accentset{\approx}{u}_{2}=\accentset{\approx}{q}_{TT}(x_{2},k_{1},k_{3},\omega)$
(18)

We now have two coupled linear differential equations 17 and 18 for the
variables $p$ and $u_{2}$. In principle one cannot be solved without also
solving the other. If, however, one of the variables, such as the velocity, is
known then only one equation has to be solved. However, the result depends on
the accuracy with which the individual variable is defined, and that is often
subject to large uncertainty, and flow specific approximations. Nevertheless
it is the basis for the Blake-TNO approach to obtain surface pressure (Blake,
2017).

##  4 The Surface Pressure Calculations based on the Orr Sommerfeld Equation

As an alternative to the formulation of the problem based on the Poisson’s
equation for the pressure, Landahl (1967) considered the problem in terms of
the inhomogeneous Orr-Sommerfeld equation. This approach treats the non-linear
terms as the effective sources of all the fluctuating quantities in the
boundary layer, and as such represents a complete solution to the Navier
Stokes equations if the non-linear terms are known. A more detailed derivation
of this approach was given by Chase (1991) who started with the vorticity
equation in a parallel shear flow and applied the wavenumber transforms
specified in §3. Then by using Squires transformation the result was reduced
to the Orr-Sommerfeld equation for two of the three components of the unsteady
velocity, with the third component related to the other by the momentum
equation. In this section we will re-visit Chase’s derivation of the
inhomogeneous Orr-Sommerfeld equation and then show how the surface pressure
fluctuations can be related to the vorticity in the flow.

###  4.1 Squires Transformation

As before, we will consider a parallel shear flow that is defined by the mean
velocity $U=(U(x_{2}),0,0)$ in the $x_{1}$ direction. Superimposed on the mean
flow there is a velocity perturbation, and for simplicity, we will only
consider a single wavenumber frequency component, which will be specified as

$u_{i}(\bm{x},t)=w_{i}(x_{2},k_{1},k_{3},\omega)e^{-i\omega
t+ik_{1}x_{1}+ik_{3}x_{3}}$ (19)

The general solution is then given by the inverse Fourier transforms of each
variable for each frequency and wavenumber. We are concerned with turbulent
flows that are statistically stationary in time and homogeneous in the plane
of the mean flow, and this requires that the frequency $\omega$ and the two
wavenumbers $k_{1}$ and $k_{3}$ are real and have values in the range
$\pm\infty$.

For convenience we introduce the wavenumber vector $\bm{k}=(k_{1},0,k_{3})$
and the unit vector in the direction of $\bm{k}$ which is
$\bm{s}=(k_{1}/k_{s},0,k_{3}/k_{s})$ where $k_{s}=(k_{1}^{2}+k_{3}^{2})^{1/2}$
(see figure 1).

Figure 1:  The coordinates and the directions of the unit vectors $\bm{s}$,
$\bm{j}$ and $\bm{t}$.

There are two vectors orthogonal to $\bm{s}$, the first is $\bm{j}$ that lies
in the $x_{2}$ direction and the second is
$\bm{t}=(-k_{3}/k_{s},0,k_{1}/k_{s})$ which lies in the plane of the flow.
Then $k_{1}x_{1}+k_{3}x_{3}=\bm{k}\cdot\bm{x}=k_{s}x_{s}$ where
$x_{s}=\bm{s}\cdot\bm{x}$.

This leads to some important simplifications for incompressible flows for
which $\nabla\cdot\bm{u}=0$. It follows that

$ik_{1}w_{1}+w_{2}^{\prime}+ik_{3}w_{3}=ik_{s}w_{s}+w_{2}^{\prime}=0$ (20)

where $w_{s}$ is the component of the velocity in the direction of the unit
vector $\bm{s}$ and we have used a prime to indicate differentiation with
respect to $x_{2}$. It follows that the $w_{s}$ and $w_{2}$ components of the
velocity are specified by the stream function $\psi(x_{2})$ such that

$\displaystyle w_{s}$ $\displaystyle=\psi^{\prime}$ (21a) $\displaystyle
w_{2}$ $\displaystyle=-ik_{s}\psi$ (21b)

and the vorticity in the direction of the unit vector $\bm{t}$ is

$\omega_{t}=k_{s}^{2}\psi-\psi^{\prime\prime}$ (22)

There is also a velocity component in the direction $\bm{t}$ which is defined
as $w_{t}$ and determines the vorticity components in the directions normal to
the flow and in the direction of the wave vector since

$\displaystyle\omega_{2}$ $\displaystyle=-ik_{s}w_{t}$ (23a)
$\displaystyle\omega_{s}$ $\displaystyle=w_{t}^{\prime}$ (23b)

To obtain $w_{t}$ we consider the momentum equation

$\dfrac{\partial u_{i}}{\partial t}+U\dfrac{\partial u_{i}}{\partial
x_{1}}+u_{2}U^{\prime}\delta_{i1}+\dfrac{1}{\rho_{0}}\dfrac{\partial
p}{\partial x_{i}}-\nu\dfrac{\partial^{2}u_{i}}{\partial
x_{j}^{2}}=-u_{j}\dfrac{\partial u_{i}}{\partial x_{j}}$ (24)

which after applying the the Fourier transforms in time and space gives

$i(k_{1}U-\omega)w_{i}+w_{2}U^{\prime}\delta_{i1}+ik_{i}p-\nu(w_{i}^{\prime\prime}-k_{s}^{2}w_{i})=-q_{i}$
(25)

Evaluating this in the $\bm{t}$ direction gives

$i(k_{1}U-\omega)w_{t}+w_{2}U^{\prime}t_{1}-\nu(w_{t}^{\prime\prime}-k_{s}^{2}w_{t})=-q_{t}$
(26)

where the pressure term has been eliminated because there is no variation of
the perturbed quantities in the $\bm{t}$ direction since it is parallel to the
wave fronts. If the non-linear terms, specified by $q_{t}$, are small, which
implies that $U>>w_{2}$ and $w_{s}$, then this gives a second order
differential equation that relates $w_{t}$ to $w_{2}$, and hence $\psi$. If
the Reynolds number of the flow is large, and the viscous terms are ignored,
then there is a simple algebraic relationship between $w_{t}$ and $w_{2}$, and
the problem is reduced to determining a single perturbation quantity such as
$w_{2}$ or $\psi$.

The representation of a three dimensional flow perturbation in a parallel
shear flow by a single stream function $\psi$ is known as Squires theorem, and
can be found in many texts including Drazin & Reid (2004).

###  4.2 The Inhomogeneous Orr Sommerfeld Equation

The vorticity equation can be obtained by taking the curl of the momentum
equation and, if the mean vorticity is $\bm{\Omega}(x_{2})=(0,0,\Omega)$ and
the vorticity perturbation is $\bm{\omega}(x,t)$, then we find that,

$\dfrac{\partial\bm{\omega}}{\partial
t}+\left(\bm{U}\cdot\nabla\right)\bm{\omega}-\left(\bm{\omega}\cdot\nabla\right)\bm{U}+\left(\bm{u}\cdot\nabla\right)\bm{\Omega}-\left(\bm{\Omega}\cdot\nabla\right)\bm{u}-\nu\nabla^{2}\bm{\omega}=\left(\bm{\omega}\cdot\nabla\right)\bm{u}-\left(\bm{u}\cdot\nabla\right)\bm{\omega}$
(27)

For the case of the parallel shear flow with a harmonic perturbation, as
discussed above, the vorticity in the $\bm{t}=(-k_{3}/k_{s},0,k_{1}/k_{s})$
direction is given by $\bm{\omega}\cdot\bm{t}=\omega_{t}(x_{2})\exp(-i\omega
t+i\bm{k}\cdot\bm{x})$ and is the solution to the equation

$ik_{1}\left(U-c\right)\omega_{t}+\omega_{2}U^{\prime}k_{3}/k_{s}+w_{2}\Omega^{\prime}k_{1}/k_{s}-ik_{3}\Omega
w_{t}-\nu\left(\omega_{t}^{\prime\prime}-k_{s}^{2}\omega_{t}\right)=\theta_{t}$
(28)

where the non linear terms are specified by $\theta_{t}$ (see Chase (1991)).
Since the mean vorticity is related to the mean flow velocity by
$\Omega=-U^{\prime}$ and, as was shown above, $\omega_{2}=-ik_{s}w_{t}$, the
second and fourth terms cancel so we obtain

$ik_{1}\left(U-c\right)\omega_{t}-w_{2}U^{\prime\prime}k_{1}/k_{s}-\nu\left(\omega_{t}^{\prime\prime}-k_{s}^{2}\omega_{t}\right)=\theta_{t}$
(29)

Noting that $w_{2}=-ik_{s}\psi$, equation 29 and equation 22 give two coupled
differential equations for the vorticity and the stream function,

$\displaystyle
ik_{1}(U-c)\omega_{t}+ik_{1}U^{\prime\prime}\psi-\nu(\omega_{t}^{\prime\prime}-k_{s}^{2}\omega_{t})$
$\displaystyle=\theta_{t}$ (30a)
$\displaystyle\psi^{\prime\prime}-k_{s}^{2}\psi$ $\displaystyle=-\omega_{t}$
(30b)

and when these are combined we obtain

$-ik_{1}\left(U-c\right)(\psi^{\prime\prime}-k_{s}^{2}\psi)+ik_{1}\psi
U^{\prime\prime}+\nu(\psi^{\prime\prime\prime\prime}-2k_{s}^{2}\psi^{\prime\prime}+k_{s}^{4}\psi)=\theta_{t}$
(31)

This is the inhomogeneous form of the Orr-Sommerfeld (OS) equation, with the
non-linear terms included on the right side as source terms. It provides the
complete solution for the flow given the relationships in the previous
section. The non-linear term is, strictly, part of the solution. However, it
is a function of perturbations at different wavenumbers, and these are
uncorrelated with the perturbations at the frequency and wavenumbers being
considered. We therefore argue that the non-linear terms are effectively
source terms for the perturbations since they do not interfere with the wave
motion that is described by the terms on the left side of the equation.

We normalize all the variables in these equations based on the outer
lengthscale $\delta$ and the outer flow speed $U_{0}$ so they take the form

$\displaystyle(U-c)\omega_{t}+U^{\prime\prime}\psi+i\epsilon(\omega_{t}^{\prime\prime}-\alpha^{2}\omega_{t})$
$\displaystyle=\theta_{t}$ (32a)
$\displaystyle\psi^{\prime\prime}-\alpha^{2}\psi$ $\displaystyle=-\omega_{t}$
(32b)

Where $\epsilon=\nu/k_{1}U_{o}\delta^{2}=1/\alpha_{1}Re$,
$\alpha_{1}=k_{1}\delta$ and $\alpha=k_{s}\delta$. Since $Re$ is the outer
flow Reynolds number we can always assume that this is a large parameter and
$\epsilon<<1$. All functions are dependent on $y=x_{2}/\delta$ and a prime
donates a differentiation with respect to non-dimensional distance from the
wall.

###  4.3 The Surface Pressure

The surface pressure $p_{s}$ is obtained from the momentum equation 25
evaluated at the wall in the $\bm{s}$ direction. Because of the no slip and
non penetration boundary condition this simplifies to

$ik_{s}p_{s}=\nu\left[(w_{s}^{\prime\prime}-k_{s}^{2}w_{s})\right]_{x_{2}=0}=-\nu\omega_{t}^{\prime}(0)$
(33)

Non-dimensionalizing this based on outer units, with pressure normalized on
$\rho U_{0}^{2},$ gives the wall pressure as

$\left(\dfrac{\rho_{0}U_{0}^{2}}{\delta}\right)i\alpha
p_{s}=-\omega_{t}^{\prime}(0)\left(\dfrac{\mu U_{0}}{\delta^{2}}\right)$ (34)

So

$p_{s}=i\epsilon\omega_{t}^{\prime}(0)=i\omega_{t}^{\prime}(0)/\alpha_{1}Re$
(35)

This gives a relatively simple set of equations 32, and 35 that can be solved
for the surface pressure providing that the non-linear terms are known. These
are specified by Chase (1991) as

$\theta_{t}=\left(\accentset{\approx}{T}_{j2}^{\prime\prime}+\alpha^{2}\accentset{\approx}{T}_{j2}\right)s_{j}+i\alpha\left(\accentset{\approx}{T}_{ij}^{\prime}s_{i}s_{j}-\accentset{\approx}{T}_{22}^{\prime})\right)$
(36)

where $\accentset{\approx}{T}_{ij}$ is the frequency space transform of
$u_{i}u_{j}/U_{o}^{2}$.

There are several different methods for solving the inhomogeneous OS equation.
Landahl (1967) used a modal approach solving for the eigenvalues of the
complex phase speed $c$ and the associated orthogonal modes that match the
boundary conditions at the surface and in the outer flow. However it was noted
that the numerical errors associated with this approach at high Reynolds
number could become large. Chase (1991) considered the asymptotic case that
the wavenumber $k_{1}$ tended to zero. He used the method of variation of
parameters to solve the inhomogeneous OS equation and was thus able to specify
a continuous solution for all wavenumbers and frequencies. However if either
of these approaches are taken the non-linear terms and the upstream boundary
conditions need to be specified to obtain a complete solution, and as was
noted above, these are difficult to both measure and model. For that reason we
will reconsider the problem by making use of the second of equation 32 which
relates the vorticity component $\omega_{t}$ to the velocity field. If this
equation can be solved in terms of the vorticity within the flow then we can
use the momentum equation 35 to obtain the surface pressure. However the
vorticity next to the surface is dominated by the viscous shear stress which
is needed to ensure the no slip boundary condition is met. This generates
vorticity at the boundary, which must extend into the flow since it forms the
viscous sublayer. In addition to this the turbulence outside the viscous
sublayer generates vorticity of it’s own, which, by the Biot Savort law must
induce a flow parallel to the surface. It is the conjecture of this paper that
the flow induced by the turbulence outside the viscous sublayer, the ‘outer’
vorticity, is exactly cancelled by the viscous shear stress at the wall, whose
gradient in the wall normal direction is exactly equal to the pressure
gradient at the wall. If this is the case then the surface pressure can be
defined simply in terms of the outer vorticity and this simplifies the
modeling and measurement of the flow characteristics that create surface
pressure fluctuations.

###  4.4 Separating the Inner and Outer Vorticity Terms

The next step in the analysis is to split the vorticity in the boundary layer
into a part that defines the surface pressure and a part that allows for the
wall shear stress. On this basis we specify

$\displaystyle\omega_{t}$ $\displaystyle=\omega_{p}+A\omega_{w}$ (37a)
$\displaystyle\psi$ $\displaystyle=\psi_{p}+A\psi_{w}$ (37b)

Where $A$ is a constant. In this equation $\omega_{p}$ is the solution to the
inhomogeneous OS equation that matches the boundary conditions that the
vorticity tends to zero far from the wall but does not impact the wall shear
stress. Hence the vorticity is zero at the wall, so $\omega_{p}(0)=0$. The
vorticity $\omega_{w}$ is the vorticity in the viscous sublayer that ensures
the no slip boundary condition, and, since the non-linear terms are zero at
the wall, it will be a solution to the homogeneous OS equation that decays at
infinity and has the boundary condition at the wall of $\omega_{w}(0)=1$. The
corresponding streamfunctions are solutions to the Poisson’s equation given by
the second of equation 32 and so can have their own boundary conditions. These
are chosen so that the stream function tends to zero at large distances from
the wall, and matches the non penetration boundary condition
$\psi(0)=\psi_{p}(0)=\psi_{w}(0)=0$. It follows then that the solutions to
equation 32 will be of the form

$\displaystyle\psi_{p}(y)$
$\displaystyle=\dfrac{1}{2\alpha}\int\displaylimits_{0_{+}}^{\infty}\omega_{p}(y_{0})\left\\{e^{-\alpha\lvert
y-y_{0}\rvert}-e^{-\alpha(y+y_{0})}\right\\}dy_{0}$ (38a)
$\displaystyle\psi_{w}(y)$
$\displaystyle=\dfrac{1}{2\alpha}\int\displaylimits_{0}^{\infty}\omega_{w}(y_{0})\left\\{e^{-\alpha\lvert
y-y_{0}\rvert}-e^{-\alpha(y+y_{0})}\right\\}dy_{0}$ (38b)

To match the no slip condition at the wall the constant $A$ is required to be

$A=-\psi_{p}^{\prime}(0)/\psi_{w}^{\prime}(0)$ (39)

and so

$A=\dfrac{1}{\psi_{w}^{\prime}(0)}\int\displaylimits_{0_{+}}^{\infty}\omega_{p}(y_{0})e^{-\alpha
y_{0}}dy_{0}$ (40)

which represents the integral of the outer vorticity in the region $y_{0}>0$
with an exponential weighting.

We can then obtain the pressure spectrum from equation 35 as

$p_{s}=i\epsilon(A\omega_{w}^{\prime}(0)+\omega_{p}^{\prime}(0))=\dfrac{i\epsilon\omega_{w}^{\prime}(0)}{\psi_{w}^{\prime}(0)}\int\displaylimits_{0_{+}}^{\infty}\omega_{p}(y_{0})e^{-\alpha
y_{0}}dy_{0}+i\epsilon\omega_{p}^{\prime}(0))$ (41)

In the viscous sublayer the mean velocity gradient is constant, and so
$U^{\prime\prime}=0$. Consequently the first equation given by equation 32 is
uncoupled from the second. Furthermore, the mean flow speed tends to zero so
we can assume, at least in the first instance when $c>>U(y)$, that
$\omega_{w}\sim\exp(-\alpha_{0}y)$ where

$\alpha_{0}^{2}=\alpha^{2}-ic/\epsilon$ (42)

Note that, for the non-dimensional frequency $\omega$, $c=\omega/\alpha_{1}$
so $c/\epsilon=(\omega/\alpha_{1})\alpha_{1}Re=\omega Re$ and we see that we
also have the relationship

$\alpha_{0}^{2}=\alpha^{2}-i\omega Re$ (43)

Since $\omega$ is of order one in typical boundary layer flows the near wall
decay rate of the shear stress vorticity is very rapid at high Reynolds
number.

The velocity perturbations induced by the shear stress at the wall can then be
assumed to be

$\displaystyle\psi_{w}(y)$
$\displaystyle=\dfrac{1}{(\alpha_{0}^{2}-\alpha^{2})}\left\\{e^{-\alpha_{0}y}-e^{-\alpha
y}\right\\}$ (44a) $\displaystyle\psi_{w}^{\prime}(y)$
$\displaystyle=\dfrac{-1}{(\alpha_{0}^{2}-\alpha^{2})}\left\\{\alpha_{0}e^{-\alpha_{0}y}-\alpha
e^{-\alpha y}\right\\}$ (44b)

and

$\omega_{w}(y)=e^{-\alpha_{0}y}$ (45)

Using these solutions gives the constants needed in equation 41 as

$\dfrac{i\epsilon\omega_{w}^{\prime}(0)}{\psi_{w}^{\prime}(0)}=\dfrac{-i\epsilon(\alpha_{0}^{2}-\alpha^{2})\alpha_{0}}{\alpha_{0}-\alpha}=\dfrac{-i\epsilon\left(\dfrac{-ic}{\epsilon}\right)\alpha_{0}}{\alpha_{0}-\alpha}$
(46)

Hence the surface pressure is

$p_{s}=-\dfrac{c\alpha_{0}}{\alpha_{0}-\alpha}\int\displaylimits_{0_{+}}^{\infty}\omega_{p}(y_{0})e^{-\alpha
y_{0}}dy_{0}+i\epsilon\omega_{p}^{\prime}(0)$ (47)

For turbulent flows at high Reynolds number where $\alpha_{0}>>\alpha$ and
$\epsilon<<1$ this simplifies to

$p_{s}=-c\int\displaylimits_{0_{+}}^{\infty}\omega_{p}(y_{0})e^{-\alpha
y_{0}}dy_{0}$ (48)

This is the key result of this paper because it shows that the surface
pressure wavenumber spectrum can be defined by the integral of the outer
vorticity as a function of distance from the wall, with an exponential
weighting factor. In obtaining this result the solution to the OS equation
next to the wall assumed that the mean flow speed was much less than the phase
speed, but this assumption is not necessary and more accurate solutions can be
obtained if needed using either Airy functions or the WKB method (see Drazin &
Reid (2004)) for a linear near wall mean velocity profile. We still, however,
need to specify the outer vorticity and so in the next section we will use a
DNS calculation of a channel flow to first verify the results given by
equation 48, and then guide the modeling of the outer vorticity. Following
that we will develop suitable models for the vorticity wavenumber spectrum
that can be used to obtain the surface pressure spectrum in terms of the flow
variables.

##  5 Verification of Results using DNS

In order to obtain the surface pressure wavenumber spectrum using equation 47
it is necessary to split the vorticity into the part defined by the wall shear
stress and the part that is caused by the non-linear source terms and upstream
boundary conditions. The vorticity associated with the wall shear stress has
been modeled as $A\omega_{w}(y)=A\exp(-\alpha_{0}y)$ and we have specified
that $\omega_{p}(0)=0$. It follows then that $A$ is equal to the vorticity at
the wall. By using a DNS calculation we can there verify that the modeling
based on equation 47 and 48 is correct by evaluating

$\omega_{p}(y)=\omega_{t}(y)-\omega_{t}(0)e^{-\alpha_{0}y}$ (49)

from the DNS calculation and then evaluating equation 48 to verify that the
pressure spectrum obtained from the DNS is reproduced. Given that, we can then
consider the properties of $\omega_{p}(y)$ to gain insights into the
mechanisms of surface pressure fluctuations in the flow.

###  5.1 The DNS Calculations

The turbulent flow data used in this work was generated using Direct Numerical
Simulation (DNS) of a periodic channel flow. The incompressible Navier-Stokes
equations were solved on a staggered grid using a second order finite
difference scheme and the second order semi-implicit Crank-Nicolson scheme
(Desjardins et al., 2008). The flow was driven in the channel by imposing a
constant pressure gradient in the streamwise direction. The friction Reynolds
number was set to be $Re_{\tau}=u_{\tau}(L_{y}/2)/\nu\approx 300$, where
$u_{\tau}=\sqrt{\tau/\rho}$ is the friction velocity and $\tau=\mu U^{\prime}$
is the surface shear stress. Periodic boundary conditions were used in the
streamwise and spanwise directions, and the no-slip boundary condition was
enforced at the top and bottom walls. A stretched Cartesian grid was used in
the wall-normal direction to resolve the viscous length scales close to the
wall. The minimum grid cell height $\Delta y$ was $0.03\delta^{+}$ next to the
wall, and the maximum $\Delta y$ was $2.4\delta^{+}$ in the core region, with
the cell height stretched using a hyperbolic tangent function. Here,
$\delta^{+}=\nu/u_{\tau}$ is the viscous length scale, $\nu=\mu/\rho$ is the
kinematic viscosity, $\mu$ is the dynamic viscosity, and $\rho$ is the fluid
density. The grid cell sizes were kept uniform in the streamwise and spanwise
directions ($\Delta x=\Delta z=3.5\delta^{+}$). Upon reaching a statistically
stationary state, the vorticity components were calculated and snapshots
separated by short time intervals of $3e$-$5t^{+}$ were saved (where
$t^{+}=\delta^{+}/u_{\tau}$ is the viscous time scale) in order to compute
time-derivatives. Additionally the wall-pressure values were stored for
computing the relevant spectra in § 5.2.

###  5.2 Surface Pressure and Vorticity from DNS

The DNS calculation provides the pressure, velocity and vorticity in the flow
as a function of space and time. In contrast, the theory given above is in the
wavenumber frequency domain and thus the DNS data must be transformed into
Fourier space as a function of $k_{1}$, $k_{3}$ and $\omega$ to give a direct
comparison with equation 48. The size of the data set for a well resolved
time-series is expected to be on the order of 10 terabytes, so to reduce the
computational effort we will only consider the surface pressure wavenumber
spectrum in space at one time instance. To obtain this, we need to take the
Fourier transform of the data with respect to $k_{1}$ and $k_{3}$, and
consider the inverse transform of equation 48 as a function of frequency, so
at time $t=0$ the surface pressure wavenumber spectrum
$p_{s}^{(t)}(\alpha_{1},\alpha_{3},t)$ is

$p_{s}^{(t)}(\alpha_{1},\alpha_{3},t)=-\int\displaylimits_{-\infty}^{\infty}c\int\displaylimits_{0_{+}}^{\infty}\omega_{p}(y_{0},\alpha_{1},\alpha_{3},\omega)e^{-\alpha
y_{0}-i\omega t}dy_{0}d\omega$ (50)

Without knowing the frequency spectrum of the vorticity this expression cannot
be evaluated exactly, but in the high Reynolds number limit we can use the
relationship that $c=\omega/\alpha_{1}$ and take the limit that
$\alpha_{0}>>\alpha$ to give

$p_{s}^{(t)}(\alpha_{1},\alpha_{3},t)=-\int\displaylimits_{-\infty}^{\infty}\dfrac{i\omega}{i\alpha_{1}}\int\displaylimits_{0_{+}}^{\infty}\omega_{p}(y_{0},\alpha_{1},\alpha_{3},\omega)e^{-\alpha
y_{0}-i\omega t}dy_{0}d\omega$ (51)

The remaining term in the inverse transform over frequency is then the outer
vorticity, which can be evaluated to give

$p_{s}^{(t)}(\alpha_{1},\alpha_{3},t)=\dfrac{1}{i\alpha_{1}}\dfrac{\partial}{\partial
t}\int\displaylimits_{0_{+}}^{\infty}\omega_{p}(y_{0},\alpha_{1},\alpha_{3},t)e^{-\alpha
y_{0}}dy_{0}$ (52)

However, to obtain the outer vorticity as a function of time we must correct
the actual vorticity using equation 49, and this includes a factor of
$\alpha_{0}$ which is frequency dependent. The inner vorticity, given by the
second term in equation 49 is therefore weighted by a factor
$\exp(-(\alpha^{2}+i\omega Re)^{1/2}y)$. This function is sharply peaked at
zero frequency, but because the fluctuating vorticity on the surface
$\omega_{t}(0,\alpha_{1},\alpha_{3},\omega)$ must be zero at zero frequency
the peak is eliminated. The inverse Fourier transform of
$\omega_{t}(0,\alpha_{1},\alpha_{3},\omega)\exp(-(\alpha^{2}+i\omega
Re)^{1/2}y)$ is then approximated by

$\omega_{t}(0,\alpha_{1},\alpha_{3},t)\exp(-(\alpha^{2}+i\omega_{m}Re)^{1/2}y)$
(53)

where $\omega_{m}$ is the peak frequency in the vorticity spectrum. This
approximation is most accurate at the wall, where $y=0$ but tends to amplify
high frequency fluctuations in the inner vorticity when $y>0$. It will be most
accurate near the spectral peak of the fluctuations where most energy is
contained. The outer vorticity in equation 52 is empirically dependent on the
choice of $\omega_{m}$ and the sensitivity to this parameter choice will be
discussed below.

Figure 2 shows the comparison between the surface pressure wavenumber spectrum
at a fixed time with the surface pressure spectrum predicted from equation 52
assuming that $\omega_{m}=3$ The result is shown for a wavenumber of $k_{3}=0$
so $\omega_{t}=\omega_{3}$ and is obtained by integrating the $\omega_{3}$
component of the vorticity over the spanwise direction $x_{3}$. The agreement
is seen to be excellent using this value of $\omega_{m}$. Increasing
$\omega_{m}$ to $4$ gives a level that is on the order $1dB$ lower than that
shown at low wavenumbers, but a better fit at high wavenumbers. Decreasing
$\omega_{m}$ to $2$ gives a level that is on the order $2dB$ higher than that
shown, with a larger error at high wavenumbers. These variations indicate that
the results are not very sensitive to the choice of $\omega_{m}$.

Figure 2:  The prediction of surface pressure spectra from DNS data using a
non-dimensional frequency of $\omega_{m}=3$.

We can also evaluate the distribution of the rms vorticity using equation 49,
which is shown in Figure 3. We conclude that the shear stress correction is
limited to less than 10 wall units from the wall. The reason that the
corrected rms value is not exactly zero at the wall is because only mesh
points inside the domain have been used.

Figure 3:  The rms of the spanwise vorticity as a function of distance from
the wall.

##  6 The Surface Pressure Spectrum

The theory given in the previous sections relates the instantaneous surface
pressure wavenumber frequency spectra to the vorticity in the flow. In most
applications we need the spectral averaged wavenumber frequency spectrum and
in this section we will consider a model for this quantity based on the
wavenumber frequency spectrum of the vorticity in the flow. We will only
consider high Reynolds number turbulent boundary layers as distinct from the
DNS of the channel flow discussed above. As a consequence we will work with
dimensional units and scales that are several order of magnitude greater than
the DNS calculations.

For a time stationary flow that is homogeneous in the $x_{1}$ and $x_{3}$
directions we can define the wavenumber surface pressure spectrum
$\Phi_{pp}(\omega,k_{1},k_{3})$ and the pressure spectrum $S_{pp}(\omega)$ at
a point using

$\displaystyle{\Phi}_{pp}(\omega,k_{1},k_{3})$
$\displaystyle=\dfrac{\pi^{3}}{R^{2}T}E\left[\lvert\tilde{p}(\omega,k_{1},k_{3})\rvert^{2}\right]$
(54a) $\displaystyle\hat{\Phi}_{pp}(\omega,k_{1})$
$\displaystyle=\int\displaylimits_{-\infty}^{\infty}{\Phi}_{pp}(\omega,k_{1},k_{3})dk_{3}$
(54b) $\displaystyle S_{pp}(\omega)$
$\displaystyle=\int\displaylimits_{-\infty}^{\infty}\hat{\Phi}_{pp}(\omega,k_{1})dk_{1}$
(54c)

Then by using equation 48 we obtain

$\Phi_{pp}(\omega,k_{1},k_{3})=\left\lvert{\rho_{0}c}\right\rvert^{2}\int\displaylimits_{0}^{\infty}\int\displaylimits_{-x_{2}}^{\infty}e^{-k_{s}(2x_{2}+z_{2})}\Omega_{tt}(\omega,k_{1},k_{3},x_{2},z_{2})dx_{2}dz_{2}$
(55)

where

$\Omega_{tt}(\omega,k_{1},k_{3},x_{2},z_{2})=\dfrac{\pi^{3}}{R^{2}T}Ex\left[\accentset{\approx}{\omega}_{t}(\omega,k_{1},k_{3},x_{2})\accentset{\approx}{\omega}_{t}^{*}(\omega,k_{1},k_{3},x_{2}+z_{2})\right]$
(56)

We can also relate the vorticity spectrum to its correlation function. If the
vorticity correlation lengthscale in the vertical direction is small then we
can assume that each correlated vortex element is convected with the local
mean flow speed and model the wavenumber spectrum as

$\displaystyle\Omega_{ij}(\omega,k_{1},k_{3},x_{2},z_{2})$
$\displaystyle=\dfrac{1}{(2\pi)^{3}}\int\displaylimits_{-\infty}^{\infty}\int\displaylimits_{-\infty}^{\infty}\int\displaylimits_{-\infty}^{\infty}W_{ij}(x_{2},z_{1}-U(x_{2})\tau,z_{2},z_{3})e^{i\omega\tau-
ik_{1}z_{1}-ik_{3}z_{3}}d\tau dz_{1}dz_{3}$ (57)
$\displaystyle=\dfrac{\delta(\omega-
k_{1}U(x_{2}))}{(2\pi)^{2}}\int\displaylimits_{-\infty}^{\infty}\int\displaylimits_{-\infty}^{\infty}W_{ij}(x_{2},z_{c},z_{2},z_{3})e^{-ik_{1}z_{c}-ik_{3}z_{3}}dz_{c}dz_{3}$
(58)

where $W_{ij}$ is the vorticity correlation function in the moving frame. It
is shown in Glegg & Devenport (2017) that measurements of boundary layer
turbulence velocity fluctuations are well modelled by homogeneous isotropic
turbulence models, such as the Von Karman or Liepmann spectrum, provided the
correct length scales are chosen. Furthermore, Batchelor (1999) shows that in
homogeneous isotropic flows the vorticity spectrum is related to the velocity
wavenumber spectrum by
$\Omega_{ij}(k_{1},k_{2},k_{3})=k^{2}\Phi_{ij}(k_{1},k_{2},k_{3})$. On this
basis we could assume that

$W_{ij}(x_{2},z_{c},z_{2},z_{3})=\int\displaylimits_{-\infty}^{\infty}\int\displaylimits_{-\infty}^{\infty}\int\displaylimits_{-\infty}^{\infty}\Phi_{ij}(x_{2},k_{1},k_{2},k_{3})k^{2}e^{ik_{1}z_{c}+ik_{2}z_{2}+ik_{3}z_{3}}dk_{1}dk_{2}dk_{3}\\\
$ (59)

and specify the turbulence wavenumber spectrum in terms of the energy
spectrum, so

$\Phi_{ij}(x_{2},k_{1},k_{2},k_{3})=\left(\delta_{ij}-\dfrac{k_{i}k_{j}}{k^{2}}\right)\dfrac{E(x_{2},k)}{4\pi
k^{2}}$ (60)

where $k^{2}=k_{1}^{2}+k_{2}^{2}+k_{3}^{2}$. However, this approach leads to
non-convergent integrals because the energy spectrum asymptotes to $1/k^{5/3}$
for a Von Karman turbulence model, and $1/k^{2}$ for a Liepmann model. For the
integrals over wavenumber to converge the energy spectrum must decay as
$1/k^{3}$ or greater. To address this issue we will define the vorticity
energy spectrum that has the characteristic

$W_{ij}(x_{2},z_{c},z_{2},z_{3})=\int\displaylimits_{-\infty}^{\infty}\int\displaylimits_{-\infty}^{\infty}\int\displaylimits_{-\infty}^{\infty}\left(\delta_{ij}-\dfrac{k_{i}k_{j}}{k^{2}}\right)\dfrac{E_{v}(x_{2},k)}{4\pi}e^{ik_{1}z_{c}+ik_{2}z_{2}+ik_{3}z_{3}}dk_{1}dk_{2}dk_{3}\\\
$ (61)

where the mean square vorticity $\omega_{rms}^{2}(x_{2})$ is given by the
integral of the vorticity energy spectrum $E_{v}(x_{2},k)$ as

$\omega_{rms}^{2}(x_{2})=2\int\displaylimits_{0}^{\infty}E_{v}(x_{2},k)k^{2}dk$
(62)

and so

$\Omega_{tt}(\omega,k_{1},k_{3},x_{2},z_{2})=\dfrac{\delta(\omega-
k_{1}U(x_{2}))}{4\pi}\int\displaylimits_{-\infty}^{\infty}E_{v}(x_{2},k)e^{ik_{2}z_{2}}dk_{2}$
(63)

and the wavenumber spectrum of the pressure is, since $c=\omega/k_{1}$

$\Phi_{pp}(\omega,k_{1},k_{3})=\dfrac{\rho^{2}}{4\pi}\int\displaylimits_{0}^{\infty}\int\displaylimits_{-x_{2}}^{\infty}\int\displaylimits_{-\infty}^{\infty}e^{-2k_{s}x_{2}}U^{2}(x_{2})E_{v}(x_{2},k)\delta(\omega-
k_{1}U(x_{2}))e^{ik_{2}z_{2}-k_{s}z_{2}}dx_{2}dz_{2}dk_{2}$ (64)

The final evaluation of this integral depends on the modeling of the vorticity
energy spectrum,which we will assume has a simple exponential decay determined
by the viscous scale $L_{v}$ which is to be determined. On this basis we
define

$E_{v}(x_{2},k)=L_{v}^{3}\omega_{rms}^{2}(x_{2})e^{-kL_{v}}$ (65)

which corresponds to a turbulent energy spectrum of $E(k)=E_{v}(k)/k^{2}$ with
an additional exponential decay to account for the small scales. However these
relationships are only orders of magnitude and it does not follow that the
turbulent kinetic energy of the turbulence is proportional to the mean square
vorticity as a function of height above the wall.

The integral over $k_{2}$ in equation 63 is now given as

$L_{v}^{3}\omega_{rms}^{2}(x_{2})\int\displaylimits_{-\infty}^{\infty}e^{ik_{2}z_{2}-kL_{v}}dk_{2}$
(66)

and in the limit that $L_{v}$ tends to zero this becomes

$2\pi L_{v}^{3}\omega_{rms}^{2}(x_{2})\delta(z_{2})$ (67)

and it follows that

$\Phi_{pp}(\omega,k_{1},k_{3})=\dfrac{\rho^{2}}{2}\int\displaylimits_{0}^{\infty}e^{-2k_{s}x_{2}}U^{2}(x_{2})L_{v}^{3}\omega_{rms}^{2}(x_{2})\delta(\omega-
k_{1}U(x_{2}))dx_{2}$ (68)

The wavenumber frequency spectrum at a point is then

$\Phi_{pp}(\omega,k_{1})=\dfrac{\rho_{0}^{2}}{2}\int\displaylimits_{-\infty}^{\infty}\left(e^{-2k_{s}y_{c}}L_{v}^{3}\omega_{rms}^{2}(y_{c})U^{2}(y_{c})/k_{1}U^{\prime}(y_{c})\right)dk_{3}$
(69)

Where $y_{c}$ is the height where $\omega=k_{1}U(y_{c})$ . The spectrum is
given by

$S_{pp}(\omega)=\rho^{2}\int\displaylimits_{0}^{\infty}\left[\int\displaylimits_{-\infty}^{\infty}e^{-2k_{s}x_{2}}L_{v}^{3}\omega_{rms}^{2}(x_{2})U(x_{2})dk_{3}\right]_{k_{1}=\omega/U(x_{2})}dx_{2}$
(70)

and the non-dimensional one sided spectrum

$\dfrac{G_{pp}(\omega)U_{o}}{(\rho_{o}u_{\tau}^{2})^{2}\delta}=\left(\dfrac{U_{o}}{u_{\tau}}\right)^{4}\int\displaylimits_{0}^{\infty}\left[\int\displaylimits_{-\infty}^{\infty}e^{-2k_{s}x_{2}}\left(\dfrac{L_{v}}{\delta}\right)^{3}\left(\dfrac{\omega_{rms}(x_{2})\delta}{U_{o}}\right)^{2}\dfrac{U(x_{2})}{U_{o}}dk_{3}\right]_{k_{1}=\omega/U(x_{2})}dx_{2}$
(71)

In order to implement this model we need to specify the distribution of the
mean square vorticity in the boundary, the mean velocity profile and the
viscous length scale $L_{v}$. The mean velocity profile and the mean square
vorticity profile is readily available from the DNS calculations, but the
viscous length scale has to be estimated. Figure 4 shows the actual
distribution of rms vorticity and a model that captures the near wall
behaviors defined as

$\omega_{rms}^{2}(x_{2})=\left(\dfrac{\pi
U_{o}^{2}}{100\delta^{2}}\right)x_{2}^{+}exp(-x_{2}^{+}/10)$ (72)

The model correctly specifies the near wall vorticity, but underestimates the
vorticity in the outer layer.

Figure 5 shows the spectrum calculated using this profile and the rms
vorticity profile from the DNS. In both cases the actual mean velocity profile
was used, and the length scale $L_{v}=11$ wall units. The results are compared
to the Goody model which is a compilation of experimental measurements made at
much higher Reynolds number and extrapolated to the Reynolds number of the
DNS. Clearly the spectral shape is well matched apart from at high frequencies
where the Goody model tends to role off faster than the prediction. The level
was found to be very sensitive to the choice of $L_{v}$, but, as will be
discussed below the choice of eleven wall units is not unreasonable.

Figure 4:  A comparison of the rms vorticity as a function of distance from
the wall, with the model defined by equation 72. (dashed line DNS calculation,
solid line model) Figure 5:  The predicted surface pressure spectrum compared
to the Goody model. Solid line is the Goody model, the dashed line is the
vorticity based model, and the dashed dotted line is the vorticity model with
the mean square vorticity distribution obtained from the DNS.

##  7 Discussion

In the previous section we have developed a modeling approach for the surface
wavenumber spectra (equations 68 and 69) and surface pressure spectra
(equation 70) in a turbulent boundary layer. Unlike previous methods, which
are based on the solution to a Poisson’s equation, the model presented here
considers the solution to the OS equation in the viscous sublayer, and relates
the surface pressure spectrum to the distribution of mean square vorticity as
a function of height from the wall, the mean flow velocity profile, and a
viscous length scale. The approach and modeling has been shown to be
consistent with empirical models and DNS calculations of the flow, and so
shows great promise for applications in more sophisticated flows that cannot
be characterized by a zero pressure gradient parallel flow. For locally
defined flows, the mean velocity profile and the mean square vorticity
distribution can probably be obtained quite accurately from lower order models
such as RANS, but this is beyond the scope of the present paper. The
characteristic that is unlikely to be obtained from measurements or lower
order models is the viscous length scale. However the DNS provides insight
into this parameter since it resolves all the small scale turbulent
structures.

In Figure 6 we show the vorticity correlation functions as a function of
distance normal to the wall for reference points that are at different heights
from the surface. The correlation is not symmetric about the reference point
as would be expected due to the presence of the wall. However the correlation
becomes increasingly symmetric at larger distances from the wall. The most
important conclusion from this figure is that the length scale is less than 10
wall units, at all heights. This scale is hard to resolve with RANS
calculations, and implies that the vorticity distribution is only locally
correlated.

The same conclusion can be drawn regarding the streamwise and spanwise
vorticity as shown in Figure 7. Only the one sided correlation functions have
been shown so that the isotropy of the flow can be assessed. These results are
for the spanwise component of the vorticity and so do not include the effect
of the streamwise vorticity that may be indicative of vortex streaks. However
the streamwise vortex streaks are not strongly coupled with the aeroacoustics
of these types of flow, such as trailing edge noise, and will only contribute
at much lower frequencies than the spanwise vorticity. Not withstanding the
importance of vortex streaks to the overall features of the flow, we see from
Figure 7 that the spanwise vorticity correlation functions are remarkably
isotropic, and that their length scales are small. Close to the wall the
streamwise length scale is significantly larger than the others (figure 7(a),
solid line), but as the reference point moves away from the wall to 22 wall
units the lengthscales are very similar, and approximately equal to the 11
wall units needed to fit the Goody spectrum. It is very unlikely that such a
small length scale could be measured accurately, which implies that to fully
understand these flows high resolution numerical methods need to be used.

Figure 6: The correlation functions of the spanwise vorticity as a function of
displacement $\Delta y$ from a reference point at a height $y$ above the
surface.

(a)

(b)

(c)

Figure 7: The correlation functions of the spanwise vorticity for reference
heights $y$ and displacements in (a) the streamwise direction, (b) the
direction normal to the wall and (c) the spanwise direction.

##  8 Conclusion

In this paper we have discussed the generation of surface pressure
fluctuations below a turbulent boundary layer. It has been shown that the
transmission of pressure through a shear flow is not fully described by a
Poisson’s equation, and to fully account for the effect of a mean shear flow
the full set of the Navier Stokes equations must be solved. We have addressed
this by considering the OS equation following the approach introduced by
Landahl (1967) and Chase (1991). By working in the wavenumber domain, it was
shown how the flow variables can be specified in terms of an effective source
term that depends on a set of source terms that are only dependent on non-
linear shear stress fluctuations in the flow. Then by separating the OS
equation into two coupled equations it was shown how the vorticity in the flow
could be used as an effective source term for the surface pressure
fluctuations if the vorticity associated with the wall shear stress was
subtracted out. By considering a DNS calculation of a channel flow it was
found that the influence of the near wall shear stress extended less than ten
wall units from the surface, and the outer vorticity could be used to closely
match the surface pressure wavenumber spectrum at a fixed instance in time
over a wide range of wavenumbers.

One of the ongoing debates in the field of Aeroacoustics is the definition of
the best ‘source term’ that generates aerodynamic sound. Lighthill, Lilley,
Howe and Goldstein have provided alternative formulations for the sound
generation by flow, but all these formulations are limited by the complexity
of the source terms and the difficulties in modeling them. In this paper we
have introduced yet another approach, which is limited to incompressible flows
and so will only apply in those situations where the sound is generated by an
edge condition or surface discontinuities. We have shown that the surface
pressure can be directly related to the vorticity in the flow outside the
viscous sublayer. The formulation relies on the Biot Savart Law to relate the
surface pressure to the vorticity, and the only approximation relates to the
separation of the vorticity in the viscous sublayer that is required to match
the no slip boundary condition. The key advantage of this formulation is that
the vorticity in the flow can be modeled, whereas the non-linear source terms
in the other acoustic analogies can be hard to measure or estimate. The
vorticity source term will depend on the upstream boundary conditions, the
vortex stretching, the mean shear lift up mechanism and the non linear vortex
break up mechanism. Each of these can be studied separately, but, as we have
shown, their combined effect is the driver of the surface pressure
perturbation.

By using an approximation to the vorticity wavenumber spectrum we have derived
a formula for the evaluation of the surface pressure spectrum, which gives a
good fit to the measured surface pressure spectrum specified by Goody (2004).
This is based on the assumption that the turbulence is locally convected by
the mean flow and that limits the eddy convection speed to less than the free
stream velocity, or the maximum flow speed in the boundary layer. This model
is quite restrictive because the primary interest in surface pressure
wavenumber spectra (Graham, 1997) is at phase speeds that exceed the mean flow
speeds in the boundary layer. The full solution, given by equation 55, will
include these phase speeds but its evaluation requires a more sophisticated
model of the vorticity distribution.

## Declaration of Interests

The authors report no conflict of interest.

## Acknowledgements

This work is supported by ONR grant number N00014-20-1-2774. The author would
like to thank Dr. Ki-Han Kim, Dr. Yin Lu Young and Dr. John Muench for their
support of this project. The work follows on from a study supported by NSF
Grant CBET-1802915. Computational resources were provided by the ONR under HPC
grant ONRDC48942624. In addition the author would like to thank Dr. William
Devenport, Nandita Hari, Matt Szoke, Alexander Gonzalez, Ignacio Jimenez,
Frank Ballestieri, Dr. William Blake and Dr. Jason Anderson for valuable
discussion on this topic and inspiring many of the ideas presented.

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