Marvin E. Goldstein and Mohamed Z. Afsar

Glenn Research Center, Cleveland, Ohio

Stewart J. Leib

Ohio Aerospace Institute, Brook Park, Ohio 

Structure of the Small Amplitude Motion on 

Transversely Sheared Mean Flows

NASA/TM—2013-217862

May 2013
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Marvin E. Goldstein and Mohamed Z. Afsar

Glenn Research Center, Cleveland, Ohio

Stewart J. Leib

Ohio Aerospace Institute, Brook Park, Ohio 

Structure of the Small Amplitude Motion on 

Transversely Sheared Mean Flows

NASA/TM—2013-217862

May 2013

National Aeronautics and

Space Administration

Glenn Research Center

Cleveland, Ohio 44135
Acknowledgments

The authors would like to thank Miss Daria Frank (Ph.D. candidate, Department of Applied Mathematics 

and Theoretical Physics, Cambridge, UK) for her excellent translation of the Möhring paper.

Available from

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Available electronically at http://www.sti.nasa.gov

Level of Review: This material has been technically reviewed by technical management. 
NASA/TM—2013-217862 1

Structure of the Small Amplitude Motion on

Transversely Sheared Mean Flows

Marvin E. Goldstein and Mohamed Z. Afsar

National Aeronautics and Space Administration

Glenn Research Center

Cleveland, Ohio 44135

Stewart J. Leib

Ohio Aerospace Institute

Brook Park, Ohio 44142

Abstract

This paper considers the small amplitude unsteady motion of an inviscid non-heat conducting 

compressible fluid on a transversely sheared mean flow. It extends a previous result given in Goldstein 

(1978(b) and 1979(a)) which shows that the hydrodynamic component of the motion is determined by 

two arbitrary convected quantities in the absence of solid surfaces or other external sources. The result is 

important because it can be used to specify appropriate boundary conditions for unsteady surface 

interaction problems on transversely sheared mean flows in the same way that the vortical component of 

the Kovasznay (1953) decomposition is used to specify these conditions for surface interaction problems 

on uniform mean flows. But unlike the Kovasznay (1953) case the arbitrary convected quantities no 

longer bear a simple relation to the physical variables. One purpose of this paper is to derive a formula 

that relates these quantities to the (physically measurable) vorticity and pressure fluctuations in the flow.

1.0 IntroductionEquation Section (Next)

The small amplitude motion of an inviscid non-heat conducting compressible fluid is governed by the 

Linearized Euler Equations, i.e., the Euler equations linearized about an arbitrary, usually steady, solution 

to those equations— customarily referred to as the base flow. The simplest case occurs when the base 

flow is completely uniform. In a now classical paper, Kovasznay (1953) showed that the unsteady 

isentropic motion on this flow can be decomposed into the sum of a purely vortical disturbance that has 

no pressure fluctuations and an irrotational disturbance that carries the pressure fluctuations. The latter 

satisfies a second-order wave equation when the flow is compressible and should, as argued by Möhring 

(1976), either decay or propagate relative to the base flow. It can, therefore be associated with the 

acoustic component of the motion on these flows. The former, which moves downstream with the mean 

flow, i.e., it is a purely convected quantity, can be associated with the remaining, hydrodynamic,

component of the motion. Any convected velocity field will satisfy the linearized momentum equation for 

this flow, but continuity only allows two of its components to be arbitrary. These two quantities can then 

be independently specified as steady state boundary conditions for unsteady surface interaction problems.

This makes the Kovasznay decomposition particularly useful for analyzing the interaction of turbulence 

(which corresponds to the hydrodynamic component of the motion) with surfaces embedded in uniform 

mean flows (Sears, 1941) or in flows that become uniform in the upstream region (Hunt, 1973; Goldstein, 

1978(a), 1979(b)). It is worth noting, however, that the Kovasznay decomposition is not unique because 

there are irrotational (homogeneous) solutions that carry no pressure fluctuations. There have been many 

attempts to extend these ideas to non-uniform base flows, but the situation is considerably more 

complicated when the entire base flow is non-uniform. The simplest case occurs when the base flow shear 

is uniform (i.e., constant). 

In 1907, Orr (1907, pp. 26-29, see Drazin and Reid, 1981, pp. 147-151) pointed out that the linearized 

incompressible vorticity equation 
NASA/TM—2013-217862 2

3

1

U 0,

y

  ∂ ∂   + ω=

∂τ ∂  

(1.1)

where τ denotes the time, y1, y2, y3 are Cartesian coordinates, with y1 being in the mean flow direction or 

equivalently the two-dimensional Rayleigh equation

2 2

3 2 2 2

11 1 1 2

UU v 0

yy y y y

∂∂ ∂ ∂ ∂ ∂ ∂    

+ ω= + + =′     ∂ ∂τ ∂ ∂τ ∂    ∂ ∂  

(1.2)

that determine the two-dimensional vorticity perturbation ω3 and the unsteady transverse velocity 

perturbation v y 2 2 ′ ( ,τ) for the two-dimensional small amplitude motion on a sheared mean flow with 

linear velocity profile

U y = λ 2 (1.3)

can be integrated to obtain 

2 2

1

2 2 2 2

1 2 1 2

, c

y v y

y y y y

  ∂∂ ∂  

+ = ω τ− ′       ∂ ∂ ∂ λ  

(1.4)

where the transverse vorticity perturbation ω3 is an arbitrary function of its arguments. Orr obtained an 

analytic solution to an initial value problem associated with this equation and used it to study the 

development of the velocity and pressure fluctuations starting from some initial state. But (1.4) can also 

be formulated as a steady-state (i.e., time-stationary) boundary value problem whose formal solution is 

given by 

( ) ( ) 1

20 2

1 2

, ,, , |

T

c

T

y v t g t y dd

x y −

∂   ′ = τ ω τ−   τ

∂ λ   ∫ ∫ x x y y (1.5)

where x = {x1, x2}, y = {y1, y2} denote the two-dimensional Cartesian coordinates, the inner integration is 

over the entire y1 – y2 plane, T denotes a large time interval and g0 denotes a Green’s function, which is 

determined by

( ) ( ) ( )

2 2

2 2 0

1 2

gt t , , |

x x

  ∂ ∂   + τ =δ −τ δ −

∂ ∂  

x y y x (1.6)

together with appropriate outgoing wave boundary conditions. 

It can be shown that v t 2

′ ( x, ) will decay very slowly with x1 when the arbitrary convected quantity ωc

(τ – y1 /λy2, y2) decays sufficiently slowly as y2 → ±∞. The latter can then be specified as an upstream 

boundary condition since(1.5) satisfies (1.2) for any choice of this quantity. Note that the statistics of any 

time-stationary turbulent shear flow can be represented by a sum of such time harmonic functions (see for 

example, Goldstein and Durbin 1980).

Equation(1.4), which bears some relation to the so-called continuous spectrum, was extended to 

three-dimensional compressible motions on base flows with constant mean shear by Möhring (1976) and 

to general planar compressible shear flows by Goldstein (1978(b), 1979(a)): hereafter referred to as G78 

and G79, respectively) who showed how this more general (non-uniform mean flow) result can used to 

formulate surface interaction problems that are relevant to aircraft noise problems in which the surfaces 
NASA/TM—2013-217862 3

are typically embedded in transversely sheared mean flows. But, unlike the uniform flow and 

incompressible constant mean shear cases, the arbitrary convected quantity ωc (τ – y1 /U(y2), y2)—or its 

equivalent—can no longer be associated with a single component of the vorticity in these more general 

flows. 

G78 and G79 obtained a result that is the natural generalization of the vortical solution in the 

Kovasznay (1953) decomposition by extending the single-convected-quantity, Orr-type, solutions to 

include an additional arbitrary convected quantity, but neither of these can be associated with a single 

component of the vorticity or, for that matter, any other specific physical (measurable) variable. 

Much of modern Rapid Distortion Theory (RDT) is concerned with the small amplitude vortical 

motion on potential flows that originate from a completely uniform upstream flow (Sagaut and Cambon, 

2008; Hunt, 1973, 1977; Goldstein, 1978(a), 1979(b)) or small scale motion on irrotational mean flows 

(e.g., Gartshore, Durbin and Hunt 1983 and Nazerenko, Kevlahan and Drubulle, 1999). An important 

application of existing rapid distortion theory is the specification of appropriate upstream boundary 

conditions, which can only be accomplished by using the Kovasznay decomposition (see, for example, 

Lele 1998, p. 432 and ff. and Colonius 2004, p.316). A natural extension of these ideas would be to 

develop a theory based on the small amplitude vortical motion on steady vortical base flows that originate 

from a parallel, or more generally, a transversely sheared mean flow in the upstream region (Colonius 

2004, pp. 323-324).

In this paper, we consider the small-amplitude motion of an inviscid non-heat-conducting 

compressible fluid on a general transversely sheared mean flow with the aim of understanding its 

underlying structure and providing the necessary formalism to extend the class of mean flows for which 

applications of RDT can be considered.

The paper begins by using a newly uncovered connection to the adjoint Rayleigh operator to extend 

the G78 and G79 results to completely general (variable density, non-planar base flow, etc.) transversely 

sheared mean flows. As in G78 and G79 the unsteady motion, which can be calculated by solving an

inhomogeneous equation, is determined by two arbitrary convected quantities in the absence of solid 

surfaces, external sources and Kelvin-Helmholtz instabilities. The relevant solution is the sum of two 

parts: a part which produces no pressure fluctuations and is driven by one of the two arbitrary convected 

quantities and a part that produces pressure fluctuations and is driven by the other arbitrary convected 

quantity. 

A relation between the convected quantities and the physical variables is worked out in Section 3.0. It 

is shown that there is a specific linear combination of these quantities that is related to a linear 

combination of the physical variables (pressure, vorticity and their derivatives) by a single algebraic 

equation, which generalizes an equation obtained by Möhring (1976) for the linear base flow profile (1.3)

with constant sound speed. But Möhring (1976) result, which is derived from the Beltrami equation

(Emanuel,1993, p. 90), does not include a second convected quantity which, as shown in G79, must be 

included in order to completely represent small amplitude vortical motion on steady vortical base flows. It 

cannot, therefore, be used to represent general three-dimensional vortical disturbances. Our results 

generalize the classical Kovasznay (1953) decomposition to arbitrary transversely sheared mean flows 

and provide the formalism needed to apply RDT to these mean flows. 

The paper is concluded with Section 4.0, which discusses the implication of these results and how 

they can be used to formulate RDT problems, i.e., problems that involve the interaction of turbulence with 

solid surfaces.

2.0 The Basic ResultEquation Section (Next)

We suppose that the flow is inviscid and non-heat conducting and assume an ideal gas so that the 

entropy is proportional to ln(p/ργ

), and the squared sound speed is γp/ρ,where p denotes the pressure, ρ

the density and γ the specific heat ratio. Then the inviscid pressure p′ = p – p0 and momentum flux
NASA/TM—2013-217862 4

, i i u v ≡ ρ ′ (2.1)

perturbations (where i v′ denotes the velocity perturbation)on a transversely sheared mean flow with 

pressure p0 = constant, velocity U(yT) and mean sound speed squared c

2

(yT) are governed by the linearized 

momentum and energy equations 

0

1 0 i

i j

j i

D u U

u p D yy

∂ ∂ +δ + =′ τ ∂∂ (2.2)

0 2 0 j

j

D p c u

D y

′ ∂

+ =

τ ∂ , (2.3)

where yT = {y2, y3}, y = {y1, y2, y3} = {y1, yT},

1

o

D

D U

τ y

∂ ∂

≡ + ∂τ ∂ (2.4)

is the convective derivative.

G79 points out that the momentum Equation (2.2) will be identically satisfied for any function φ and 

any purely convected function θ(τ – (y1/U), yT) when ui and p′ are determined by 

3

0

3

D

p D

φ ′ = − τ

(2.5)

0 1

1 2

1 , i ij i j ijk T

j j k

D U U y u

D y c yy U

  ∂ ∂∂   = δ −δ Λ +ε   θ τ−   τ ∂ ∂ ∂    

y (2.6)

where δij denotes the Kronecker delta, εijk denotes the alternating tensor and

0

1

2 j

j j

D U

yD y y

∂ ∂ ∂φ φ Λ≡ +

∂ τ ∂∂ (2.7)

denotes a kind of generalized particle displacement. 

The arbitrary function φ can then be adjusted to ensure that the energy Equation (2.3) is also satisfied by 

substituting (2.5)--(2.7) into (2.3) to obtain

3

00 0 2

3

1

2 0,

ii i

DD D U

c

D y yD y y D

  ∂ ∂ ∂ ∂φ   φ φ     + −=

τ∂ ∂ τ ∂ ∂   τ  

(2.8)

which can then be integrated to obtain

1 , ac T

y L

U

  φ = −ω τ −      y (2.9)

where
NASA/TM—2013-217862 5

3

0 0 2

3

1

2 a

ii i

D D U L c

D y yD y y

∂ ∂ ∂∂  

≡− +  

τ ∂ ∂ τ ∂∂  

(2.10)

It is well known that the momentum flux perturbation, ui can be eliminated between (2.2) and (2.3) to 

show that the pressure fluctuation p′ also satisfies Rayleigh’s equation 

L p′ = 0 (2.11)

where 

2

2 2

2

1

2 o o

i i j j

D D U L c c

D y y yy y D

  ∂ ∂ ∂∂ ∂ ≡ −−   τ∂ ∂ ∂ ∂ ∂ τ  

(2.12)

denotes the usual Rayleigh operator. But it is readily verified that 

2 0 0

1

2 2

2 2 0 0

2 2

1

2

2

a

ii i i

o oo

i i j j

D v Uv u D v vLu uL v c u u

y yD y y yD

D u DuDv Dv Du U u

v c u v c v

D y y DD D D y y y

∂ ∂ ∂∂ ∂  

−= + −   ∂ ∂ τ ∂∂ ∂ τ  

  ∂ ∂ ∂ ∂∂  

+ + −− −     τ∂ ∂ τ τ τ τ ∂ ∂ ∂    

(2.13)

for any functions u, v, which shows that the Rayleigh operator L is adjoint to La (Morse and Feshbach, 

1953, p. 870). 

Let g(y,τx,t)

Lg t ,, ( y x yx τ =δ − δ τ− | ) ( ) ( t) (2.14)

be a Green’s function for the Rayleigh operator L which exhibits outgoing wave behaviour as y→ ∞, 

where, as usual, the first two arguments denote the dependent variables and the second two denote the 

source variables. We expect g(y,τx,t) to vanish as τ → ∞ for all finite y and as y1 → ±∞ for all finite τ

when the base flow U is globally stable, and causality should ensure that it vanishes as τ → –∞. We 

therefore suppose that

( ) ( ) 1

lim , , , lim , , 0, | |

y

g tg t

τ→±∞ →±∞

yx yx τ τ= (2.15)

Setting u equal to g(y,τx,t) in (2.13), letting v be a solution to (2.9) and using the divergence theorem

shows that

( ) ( ) ( )

( ) ( ) ( )

1

2 0

, ,, , 

, , ,, ˆ

|

| |

T

c T

T

T V

T

i i

i

T S

dS d

y t g t d d

U

g t D

nc g t y D

−

−

τ

  φ =− τ ω τ−   τ

 

  ∂ τ φ +   τ Λ−

∂ τ  

∫ ∫

∫ ∫



y

x yx y y

y

y x

y x

(2.16)

where T denotes a very large but finite time interval, V is a region of space bounded by cylindrical 

surface(s) S, n =ˆ {nˆi} is the unit outward-drawn normal to S, where we have omitted terms that are 
NASA/TM—2013-217862 6

negligibly small as T → ±∞ and assumed that the contribution from the end caps can be neglected (Tam 

and Auriault, 1998). This formula expresses the solution to Equation (2.9) in terms of the volume source 

distribution ωc  (τ – y1/U(yT), yT) and the values of φ on some arbitrary cylindrical surfaces S. The Green’s 

function g is not uniquely determined by (2.14) and can be required to satisfy certain boundary conditions 

on a portion of the surface S. The present analysis is somewhat unconventional in that the direct Green’s 

function g now plays the role of an adjoint Green’s function for the solution φ.

When the surface S is at infinity, i.e., V is all space

( ) ( ) ( )

1 , ,, | , 

T

c T

T

T V

y t g t d d

U −

  φ =− τ ω τ−   τ

  ∫ ∫ x yx  y y

y

(2.17)

(which generalizes Equation (2.36) in G79) and Equation (2.6) then shows that the transverse velocity 

perturbation v⊥

′ is given by

( ) ( ) ( )

1 / , / ,, | , 

T

i

i ic T

i T

T V

U U x y v u t U g t d d

x U U ⊥

−

∂ ∂ ∂  

ρ ≡ ′ ∇ =− τ ω τ−   τ

∂ ∇   ∫ ∫ x y x  y y

y

(2.18)

with

( ) ( ) 0 0

1

, , | | 2 , ,, i

i i

D D U

g t g t Dt x Dt x x

  ∂ ∂∂

τ≡ + τ   ∂ ∂∂  y x y x (2.19)

In the more general case, where the surface integral does not vanish, Equation (2.5) shows that the 

pressure perturbation p′(x,t) is given by

( ) ( )

( )

( ) ( ) ( )

3

0 1

3

3 3

2 0 0 0

3 3

, , , , 

, , , , ˆ

|

| |

T

c T

T

T V

T

i i

i

T S

dS d

Dg t y p t d d

Dt U

Dg t Dg t D

n c

Dt y D Dt

−

−

τ

τ   ′ = ω τ−   τ

 

  τ τ ∂   φ −   Λ −   ∂ τ    

∫ ∫

∫ ∫



y

y x

x y y

y

y x y x

(2.20)

which reduces to

( ) ( )

( )

3

0 1

3

, , , , |

T

c T

T

T V

Dg t y p t d d

Dt U −

τ   ′ = ω τ−   τ

  ∫ ∫  y x

x y y

y

(2.21)

when the surface integral vanishes, i.e., when g and φ satisfy appropriate boundary conditions on any 

cylindrical surfaces that may be present in the flow. 

2.1 Interpretation of Results

Equation (2.18) can be shown to generalize the Orr result (1.5) by inserting Equation (B.12) into that 

result, noting that the integral over the second term vanishes and that the Green’s function (1.6) is selfadjoint (i.e., g0(y,τx,t) = g0(y,τx,τ)) to show that it reduces to (1.5) for two dimensional incompressible 
NASA/TM—2013-217862 7

flows with constant mean shear when the arbitrary convected quantity ωc  (τ – y1/U(yT), yT) is replace by 

the renormalized quantity

( ) ( )

1 1 2 , ,/ c T c T

T T

y y U c

U U

  

ω τ−    ≡ω τ− ∇

  

y y  y y

(2.22)

which has dimensions of vorticity (based on the rescaled velocity ui). The most significant difference

between these results is that the convected quantity ωc is no longer equal to the vorticity.

Möhring (1976) pointed out that the velocity fluctuations on a uniform base flow satisfy a third order 

wave equation which can be integrated to obtain an inhomogeneous second order equation whose 

homogeneous solutions can then be identified with the acoustic motion. But in the present case the 

acoustic motion must be identified with the homogeneous solutions of the third-order wave Equation (2.9)

, which cannot, in general, be reduced to second order because these solutions have both acoustic and 

hydrodynamic components. It can, however, be reduced to second order when the mean flow does not 

support hydrodynamic motion. This can be done for uniform mean flows by introducing the dependent 

variable Φ ≡ D0φ/Dτ (which can then be used in (2.5)and (2.6) to calculate the pressure and velocity). A 

similar reduction can also be done for two-dimensional flows with constant c

2 and mean shear (see

Möhring, 1976). This can be seen by differentiating (2.9) with respect to y2 and using (2.7) and (2.10) to 

obtain

2 2 3

0 0 2 2 1

22 2 2 2 2

2 12 2

, c

D D y cc y D y yy Dy U

∂∂ ∂   φ   Λ− Λ− =− ω τ−     τ ∂ ∂∂   τ ∂    (2.23)

which becomes upon using (2.6) to eliminate Λ2

4 2 2 3 2 2

0 22 2 0 0 0 1 1

42 2 2 2 2 1 2

1 2 2 1 2 2

1 1 , c

D D D Du y c c c u y

y D y D yy D D y y U

∂∂ ∂   φφ φ       ∂ ∂     −   −   − − =− ω τ−     λ ∂   τ ∂ τ ∂∂   τ λ τ ∂ ∂        

 (2.24)

And using (2.8) to eliminate 4 4

0 D D φ τ / leads to Möhring’s (1976) result 

2 2 2

0 1 1

2 2 22 2 1 22

1 2 2 2

1 , , c c

D y y u yy y y cD c yU yU

    ∂ ∂ λ ∂   ∂     + − =− ω τ− =− ω τ−     ∂∂ τ ∂ ∂      

 (2.25)

Möhring (1976) also showed that a similar result can be obtained for the three dimensional motion on 

two-dimensional base flows with constant c

2 and mean shear.

The unsteady motion is given by (2.5) to (2.7) with φ determined by the inhomogeneous solution

(2.17) of (2.9) for the general transversely sheared mean flow considered in this paper. In the absence of 

solid surfaces and other external sources the resulting unsteady flow consists entirely of subsonically 

propagating disturbances when the flow is subsonic and, therefore, cannot radiate to the far field when the 

flow is unbounded (Goldstein, 2005 and 2009). This can easily be verified in any particular case by 

working out the relevant far field expansion. It is therefore appropriate to identify it with the 

hydrodynamic component of the motion. As in the Kovasznay result this flow is itself the sum of two

parts, but in this case one of the parts (the part that is driven by the arbitrary convected quantity

ωc  (τ – y1/U, yT)) produces pressure fluctuations while the other part, which is driven by the arbitrary

convected quantity θ(τ – y1/U, yT),does not. But unlike the Kovasznay result, the two arbitrary convected 

quantities are not experimentally observable variables.
NASA/TM—2013-217862 8

3.0 Relation Between the Physical Variables and the Convected Quantities 

Equation Section (Next)

Formulations that represent upstream boundary conditions in terms of the arbitrary convected 

quantities, ωc  (τ – y1/U, yT) and θ(τ – y1/U, yT), are most useful when these quantities can be related to the 

physical (preferably measurable) flow variables. This section shows that there is a local (i.e., non-integral) 

equation that relates a linear combination of ωc  (τ – y1/U, yT) and θ(τ – y1/U, yT) to these variables. The 

results are obtained by using relations derived in Appendix A. Relations between the vorticity and 

pressure fluctuations for the combined acoustic-like and discrete hydrodynamic component of the 

unsteady motion are also derived.

Equations (A.8) and (A.12) show that

( ) ( ) 1

i c ijk j j k

i i

N p N

y yy

∞

⊥

∂ ∂   ∂   ω − − η =− ε ω −ω ′   ∂ ∂∂      

 (3.1)

and 

( ) ( ) 1 0 i i i ij

j

N

y

∞ ⊥   ∂η   ω −ω +ε =

∂  

(3.2)

where ωc  is related to the rescaled vortical like quantity ωc by (2.22), 

2

2 , i

i

c U N

U y

∂ ≡ ∇ ∂ (3.3)

k

i ijk

j

u

y

∂

ω ≡ε

∂ (3.4)

is a density weighted vorticity based on ui, 

( ) ( )

( )

( )

2 2

2 2

1 1

1 1 ,

,

i knm ijk in jm im jn

j nm j nm

j i j ji j i j

U U

y yy c c y yy

U U U

y yy yy y c c y y

∞ ∂ ∂ ∂θ ∂ ∂ ∂θ

ω ≡ε ε = δ δ −δ δ

∂ ∂∂ ∂ ∂∂

    ∂ ∂ ∂θ ∂ ∂θ ∂ ∂ θ = −=    

∂ ∂∂ ∂∂ ∂   ∂    

(3.5)

which has zero divergence, is the vorticity component generated by the second convected quantity θ, 

i

i

U

y

⊥

∂ η≡ Λ ∂ (3.6)

can be interpreted as a transverse particle displacement since it follows from (2.6) that

0

i

i

U D

u

y D ⊥

∂

= η ∂ τ (3.7)

and 
NASA/TM—2013-217862 9

( )

( )

,

,i j i j ji

U V UV UV

y y yy yy

∂ ∂∂ ∂∂ ≡ − ∂ ∂∂ ∂∂ (3.8)

denotes the Jacobian. Notice that

( )

( )

0 , 0 ,i j

D U

D y y

∂ θ = τ ∂ (3.9)

i.e., that ∂ θ∂ (U yy ,/ , ) ( i j) is also a convected quantity. And it follows from (3.5) that

0 ( )

2

1

1

i

j i j ji

D UU U

D y yy yy y c

∞ ∂ ∂ ∂ ∂ ∂ ∂θ  

ω =−   − τ ∂ ∂∂ ∂∂ ∂  

(3.10)

and, therefore that 

0 ( ) 0 i

i

D U

D y

  ∂ ∞

  ω =

τ ∂  

(3.11)

Equations (3.1) and (3.5) relate the arbitrary convected quantities ωc  (τ – y1/U(yT), yT) and 

θ(τ – y1/U(yT), yT) to the pressure p′, the transverse vorticity components εijkωjNk and the particle 

displacement η⊥,while Equations (3.2) and (3.5) relate the arbitrary convected quantity θ(τ – y1/U(yT), yT)

to the pressure p′,the normal vorticity component ωjNj and the particle displacement η⊥. But the particle 

displacement η⊥ (which, unlike the pressure and vorticity components, is not a physical variable in the 

usual sense) can be eliminated between Equations (3.1) and (3.2) when ∂Ni/∂yi ≠ 0 to obtain the following 

equation

( ( ) )

1

1 1

1 1

/ , k mn i c T T ij l nm

i jk l

U p N N y U

y yy y y y

  −   ∂∂ ∂     ∂ ′ ∂ ω ∂ω 

χ τ− = ε      − ε  − ∂ ∂∂ ∂ ∂ ∂           y y (3.12)

that relates the linear combination

( ( ) ) ( ) ( ) 1

1 1

1 1

/ , k c mn i c T T ij l nm

i jk l

U N N y U

y yy y y y

    − ∞ ∞   ∂ ∂ω ∂ ω ∂ω   ∂ ∂  

χ τ− ≡ ε        − ε − ∂ ∂∂ ∂ ∂ ∂            

 y y (3.13)

of the two arbitrary convected quantities ωc  (τ – y1/U(yT), yT) and θ(τ – y1/U(yT), yT) to the physical 

variables p′ and ωi. Equations (3.3), (3.10), (3.11) and (A.7) show that χc(τ – y1/U(yT), yT) is itself an 

arbitrary convected quantity.

3.1 Application to Planar Mean Flows

These results are fairly complex, but most of the relevant literature is concerned with the small 

amplitude motion on planar base flows, where c

2 and U depend on a single Cartesian coordinate (say y2). 

Equations (3.1), (3.2), and (3.5) then become much simpler and reduce to 
NASA/TM—2013-217862 10

2 1 3

2

1 2 1 3

c

dN p N

y dy y y

⊥

∂     ∂ω ∂ω

ω− − η = − ′     ∂ ∂ ∂    

 (3.14)

2

2

2 3

1

N y

⊥ ∂η ∇ θ=ω +

∂  (3.15)

where

2 2

2

2 2

1 3 y y

∂ ∂ ∇≡ +

∂ ∂  (3.16)

These equations show that the arbitrary convected quantities ωc  (τ – y1/U(yT), yT) and θ(τ – y1/U(yT), yT)

are now related to p′ the vorticity components ω1, ω3 and the particle displacement η⊥ by fairly simple 

relations. Equations (3.12) and (3.13) now simplify to

( ( ) ) ( ) 2

1 2 22 2 1

13 2

/ , c T

p y Uy N N

yy y

∂∂ ∂   ′ χ τ− = − ω − ∇ ω   ∂∂ ∂     y (3.17)

where 

( ( ) ) 2 2

1 2

1 3 22

1 / , , c c T

dN y Uy y y N dy

∂   ∂ω

χ τ− ≡ − ∇ θ   ∂ ∂   

  y (3.18)

and we have used the fact that the solenoidal vectors ωi and ( )

i

∞ ω have zero divergence.

Notice that Equation (3.14) is now independent of ( )

i

∞ ω and, therefore of the second convected 

quantity θ. But the divergence ∂Ni/∂yi is equal to zero for the constant shear-constant c

2 parallel mean 

flow (1.3), since Ni is a is constant in that case and it follows from (2.22) that Equation (3.14) then 

reduces to Möhring’s (1976) result

3 1

2

1 1 3

c

p

y yy c

∂ λ   ′ ∂ω ∂ω   ω− = −

∂ ∂∂   (3.19)

that relates the single convected variable ωc to the physical variables. Equation (3.17) relates the 

combined convected quantity χc  (τ – y1/U(y2), yT) to the physical variables p′ and ωi.

4.0 Concluding Remarks

This paper is based on a formal solution to the linearized Euler equations for a transversely sheared 

mean flow which shows that the general solution to these equations is given by an inhomogeneous

solution to the adjoint Equation (2.9) and, like the Kovasznay decomposition for the unsteady motion on 

uniform flows, involves two arbitrary convected quantities θ(τ – y1/U, yT) and ωc  (τ – y1/U, yT) that can 

be used to specify appropriate boundary conditions for unsteady surface interaction problems and, 

thereby, extend the class of flows for which this can be done to arbitrary transversely sheared mean flows.

It is shown that this equation reduces to the second order wave equation derived by Möhring (1976) for 

two dimensional motions on the linear base flow (1.3) with constant sound speed once an appropriate 

dependent variable is introduced. But unlike the Kovasznay result, the hydrodynamic component carries 

pressure fluctuations and the acoustic component is not irrotational. Section 3.0 shows that there is a 
NASA/TM—2013-217862 11

linear combination of the convected quantities that is equal to a linear combination of the pressure and 

vorticity

The results developed in this paper are expected to be particularly useful for representing incident 

vortical disturbances in Rapid Distortion Theory problems involving the interaction of turbulence with 

surfaces embedded in transversely sheared base flows or, more generally, in vortical base flows that 

asymptote to transversely sheared mean flows in the upstream region. Equations (2.18), (2.21), and 

similar equations that can be derived for the remaining velocity components, determine the velocity and 

pressure fluctuations in terms of the arbitrary convected quantities ωc  (τ – y1/U, yT) and θ(τ – y1/U, yT) in 

these flows.

One such application currently under investigation is to the noise generated by the interaction of a 

high Reynolds number rectangular jet with a nearby flat plate (Afsar, Goldstein and Leib 2013). This 

work applies the formalism developed in the present paper to predict the acoustic spectrum of the noise 

scattered from the trailing edge of the plate and compares it with experimental data. The predictions 

require a model for the autocovariance of the convected quantity ωc  (τ – y1/U, yT), whose validity can be 

tested by using the relations developed in Section 3.0 to relate it to physical quantities that can be

measured in the laboratory.

NASA/TM—2013-217862 13

Appendix A.—Relation Between Convected Quantities and Physical Variables

Equation Chapter (Next) Section 1

Equations (2.5) to (2.7), (2.9)and (2.10) imply that the convected quantity ωc  is related to the pressure 

perturbation p′ and the generalized particle displacement Λi by

2

c i

i

p c

y

∂ ω= + Λ ′ ∂  (A.1)

0 1

1 2

1 , i i i ijk T

j k

D U y u

D c yy U ⊥

∂ ∂   = Λ −δ η +ε θ τ−   τ ∂ ∂   y (A.2)

where is η⊥ given by (3.6).

The density-weighted vorticity ωi is therefore, given by

0

1 2

1

kk k 1

i ijk ijk ijk ij knm ijk

j j j j j nm

u D U U

y D y y y y y yy c

⊥ ∂ ∂Λ ∂Λ ∂ ∂η ∂ ∂ ∂θ

ω ≡ε = ε +ε −ε +ε ε

∂ τ ∂ ∂ ∂ ∂ ∂ ∂∂ (A.3)

But since Equation (2.7) and the identities 

2 2

0 ijk ijk

jk jk

U

yy yy

∂ ∂φ ε =ε =

∂∂ ∂∂ (A.4)

show that

2

1

2 k

ijk ijk

j kj

U

y y yy

∂Λ ∂ ∂φ ε =ε

∂ ∂ ∂∂ (A.5)

this can be written as 

0 ( )

1

1

2 i ij ijk k i

j j k

U D

y yy Dy

∂η⊥ ∂ ∂   ∂φ ∞ ω = −ε + ε Λ − + ω   ∂ ∂ ∂ τ∂  

(A.6)

where ( )

i

∞ ω is given by (3.5). The definition (2.7) and the identity 

0 ijk

j k

U U

y y

∂ ∂

ε =

∂ ∂ (A.7)

can now be used to show that (A.6) can be written as

( )

1

1

k

i ij ijk i

j j

U

y yy

∂η⊥ ∂ ∂Λ ∞ ω = −ε − ε + ω

∂ ∂∂ (A.8)

And it, therefore, follows from (3.6) that the cross product / ε ω∂ ∂ ijk j k U y can be written as
NASA/TM—2013-217862 14

( )

( ) ( ) ( )

( )

1 n

1

1 1

1

1

1 1

n ijk j ijk jn ijk jm ijk j

k n k m k k

n i kn in k im kn in km ijk j

n k m k k

i

i ijk j

k k i kk k

U U UU U

y yy y yy y

U UU U

y y y yy y

U U UU U

yy yy yy y y

⊥ ∞

⊥ ∞

⊥ ⊥ ∞

∂ ∂ ∂∂ ∂ ∂η ∂Λ

ε ω = −ε ε − ε ε +ε ω

∂ ∂∂ ∂ ∂∂ ∂

∂η ∂ ∂∂ ∂ ∂Λ =− δ δ −δ δ + δ δ −δ δ +ε ω

∂ ∂ ∂ ∂∂ ∂

∂η ∂η ∂ ∂ ∂∂ ∂Λ ∂ = −δ + − + ε ω

∂ ∂ ∂∂ ∂∂ ∂ ∂

(A.9)

which can be solved for 1 / i ∂Λ ∂y to obtain

( ) ( ) 2

1

1 1

i

i ijk j j

i k k k

UU U U

y yy yy y

∂Λ   ∂∂ ∂ ∂η ∂η ⊥ ⊥ ∞ = −δ −ε ω −ω ∇   ∂ ∂∂ ∂∂ ∂  

(A.10)

Inserting this into (A.1) shows that

( ) ( ) 2

2 1

1 1 1

c i ijk j j

i i k k k

cU U U

p

y y y yy yy U y

⊥ ⊥ ∞   ∂ω ∂∂ ∂ ∂     ∂η ∂η ∂

= +′     −δ −ε ω −ω

∂ ∂ ∂ ∂∂ ∂∂ ∇ ∂      

 (A.11)

and, therefore, that

( ) 222

222

1 1 1

c ijk j ijk j

i i i k i k

U c cU cU

p

y y yy y y y y y UUU

⊥ ∞        ∂ω ∂ ∂∂ ∂η ∂ ∂ ∂ ∂

= +′        −ε ω +ε ω

∂ ∂ ∂∂ ∂ ∂ ∂ ∂ ∂        ∇∇∇  

 (A.12)

or

( ) ( ) 2 2

2 2

1 1

c ijk j j

i i i k

U c c U

p

y y yy U U y y

∞

⊥

    ∂ω ∂ ∂∂ ∂ ∂

= +η   ′   − ε   ω −ω

∂ ∂ ∂∂     ∇ ∇ ∂   ∂    

 (A.13)

where the identity (A.7) has again been used. 
NASA/TM—2013-217862 15

Appendix B.—Limiting Form of Equation (2.18) for Linear Base Flows

Equation Section (Next)

The easiest way to show that Equation (2.18) reduces to Equation (1.5) is by noting that the Fourier 

transform

( ) ( )

( ) ( ) ( ) ( ) ( ) 11 1

1 2 1 1

1

: , , 2| | iT T i

t kx y G k g tdx y dti

e ∞ ∞      

−∞ −∞

ω −τ − −

ω ≡ τ − −τ

π ∫ ∫ y x y, x (B.1)

of gi(y,τx,t) is related to the reduced Green’s function G k ( y x T T | : , ω 1 )

( ) ( )

( ) ( ) ( ) ( ) ( )

3

11 1 0

1 2 3 1 1

1 , : ,

2

| | T T

t kx y Dg t G k dx y dt Dti

e ∞ ∞      

−∞ −∞

ω −τ − − τ

ω ≡ − −τ

π ∫ ∫ y, x

y x (B.2)

which satisfies the reduced Rayleigh equation

( ) ( ) ( ) ( )

2 2 2

1

2 22

1 1

1 1

2

T

R T T T

c G ck L G G

Uk Uk

   ∇ ≡ ∇    + − = δ −

ω − ω − π   

 x y (B.3)

with U = U(yT) and∇ ≡∂∂ ∂∂ T { / ,/ y y 2 3} by

( ) ( ) 1 1 ( )

1

1 | | : , :, , iT T T T

T i

G k G k

ik U i x

∂

ω ≡ ω  − ω ∂  y x y x

x

(B.4)

since g(y,τx,t) depends on τ, t and y1, x1 only in the combinations t – τ and x1 – y1, respectively.

It, therefore, follows that

( ) ( ) ( )

( ) ( )

2 1 1

1 2 2

1

1 2 2

1

: , : ,

1

: ,

| |

|

T T T T

T T

G k Gk

ik U x i x

G k

ik U x i x

∂

ω = ω  − ω ∂  

∂ = ω  − ω ∂  

y x y x

x y (B.5)

and

( ) ( )

( )

( ) ( )

2 2

2 2 1

1 1 22 2

2 2 1 3

1

2

T T

G G Uk

Uk k G

x x Uk x c

∂ ∂∂   ω −

ω −   + − = δ −

∂ ∂   ω − ∂ π  

x y (B.6)

for planar base flows with constant sound speed—which means that 
NASA/TM—2013-217862 16

( ) ( ) ( )

( ) ( ) ( )

( ) ( )

2

2 2 2 2

1 1 2 1 2

2 21 3

2

2 1

2

2

2 2 2

1 1 2 1 2 1 2

2 2 3

1

2

T T

iG G Uk i Uk k G

x x Uk x

Uk

c x

G G i Uk k G i Uk k G

x x x

∂ ∂        ∂  ω −   + ω− −   ∂   ∂ ω−   ∂     

− ∂ = ω− δ −

π ∂

∂   ∂ ∂  = ω− +λ + ω− −     ∂ ∂   ∂  

x y (B.7)

and, therefore that

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

2 2

2 2

2 2 1 2 2 1

2 3 2 1

2 1 1

2

2

2

T T

TT TT

i k G Uk

x x c Uk x

i Uk k

c x

  ∂ ∂ ∂   + − = ω− δ −

∂ ∂ π ω− ∂  

  ∂ = ω− δ − −λ δ −  

π ∂ 

x y

xy xy

(B.8)

for the linear profile (1.3). And it follows from (B.8) that

( ( 21 0 ) ) 2

2 10 2

2

/ i Uy kG G ik G c

x c

∂   ω −

= −λ +   ∂  

(B.9)

where

( ) ( )

( ) ( ) ( ) ( ) ( ) 11 1

0 1 2 0 1 1

1

: , , 2| | T T

t kx y G k g tdx y dti

e ∞ ∞      

−∞ −∞

ω −τ − −

ω ≡ τ − −τ

π ∫ ∫ y x y, x (B.10)

satisfies 

( ) ( )

2 2

2

2 2 1 0 2

2 3

1

2

T T k G

x x

  ∂ ∂   + − = δ−

  ∂ ∂ π

x y (B.11)

Taking the inverse Fourier transform of (B.9) shows that 

( ) ( ) ( ) 0

20 0 2 2

1 2

1 , , || | , , , ,, D

gt gt gt

c c x x D

λ ∂ ∂

τ =− τ + τ

∂ ∂τ y x y x y x (B.12)

while the limiting form of Equation(B.11) for a completely two dimension flow is just the Fourier 

transform of Equation (1.6).
NASA/TM—2013-217862 17

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PLEASE DO NOT RETURN YOUR FORM TO THE ABOVE ADDRESS. 

1. REPORT DATE (DD-MM-YYYY)

01-05-2013 

2. REPORT TYPE

Technical Memorandum 

3. DATES COVERED (From - To)

4. TITLE AND SUBTITLE

Structure of the Small Amplitude Motion on Transversely Sheared Mean Flows 

5a. CONTRACT NUMBER

5b. GRANT NUMBER

5c. PROGRAM ELEMENT NUMBER

6. AUTHOR(S)

Goldstein, Marvin, E.; Afsar, Mohamed, Z.; Leib, Stewart, J. 

5d. PROJECT NUMBER

5e. TASK NUMBER

5f. WORK UNIT NUMBER

WBS 432938.11.01.03.02.02.14 

7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES)

National Aeronautics and Space Administration 

John H. Glenn Research Center at Lewis Field 

Cleveland, Ohio 44135-3191 

8. PERFORMING ORGANIZATION

 REPORT NUMBER

E-18655 

9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES)

National Aeronautics and Space Administration 

Washington, DC 20546-0001 

10. SPONSORING/MONITOR'S

 ACRONYM(S)

NASA 

11. SPONSORING/MONITORING

 REPORT NUMBER

NASA/TM-2013-217862 

12. DISTRIBUTION/AVAILABILITY STATEMENT

Unclassified-Unlimited 

Subject Categories: 34 and 01 

Available electronically at http://www.sti.nasa.gov 

This publication is available from the NASA Center for AeroSpace Information, 443-757-5802

13. SUPPLEMENTARY NOTES

14. ABSTRACT

This paper considers the small amplitude unsteady motion of an inviscid non-heat conducting compressible fluid on a transversely sheared 

mean flow. It extends a previous result which shows that the hydrodynamic component of the motion is determined by two arbitrary 

convected quantities in the absence of solid surfaces or other external sources. The result is important because it can be used to specify 

appropriate boundary conditions for unsteady surface interaction problems on transversely sheared mean flows in the same way that the 

vortical component of the Kovasznay decomposition is used to specify these conditions for surface interaction problems on uniform mean 

flows. But unlike the Kovasznay case the arbitrary convected quantities no longer bear a simple relation to the physical variables. One 

purpose of this paper is to derive a formula that relates these quantities to the (physically measurable) vorticity and pressure fluctuations in 

the flow. 

15. SUBJECT TERMS

Turbulence inviscid flow 

16. SECURITY CLASSIFICATION OF: 17. LIMITATION OF

 ABSTRACT

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18. NUMBER

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24 

19a. NAME OF RESPONSIBLE PERSON

STI Help Desk (email:help@sti.nasa.gov) 

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443-757-5802 

Standard Form 298 (Rev. 8-98)

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