William J. Devenport and Nicolas Spitz

Virginia Polytechnic Institute and State University, Blacksburg, Virginia

Predicting Modes of the Unsteady Vorticity

Field Near the Trailing Edge of a Blade

NASA/CR—2003-212099

March 2003

https://ntrs.nasa.gov/search.jsp?R=20030022774 2018-05-15T15:49:53+00:00Z
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William J. Devenport and Nicolas Spitz

Virginia Polytechnic Institute and State University , Blacksburg, Virginia

Predicting Modes of the Unsteady Vorticity

Field Near the Trailing Edge of a Blade

NASA/CR—2003-212099

March 2003

National Aeronautics and

Space Administration

Glenn Research Center

Prepared under Grant NAG3–2710
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PREDICTING MODES OF THE UNSTEADY VORTICITY FIELD 

NEAR THE TRAILING EDGE OF A BLADE 

EXECUTIVE SUMMARY

 Progress on predicting modes of the unsteady velocity/vorticity field of a turbulent 

boundary layer from Reynolds stress statistics is described. Prediction of these modes, that 

provide the source terms for trailing edge noise predictions in aircraft engine fans and other 

configurations, will allow for the first time detailed viscous flow effects to be included in such 

noise calculations. The key accomplishments of this work in FY02 are 

1. The development of a Matlab code for the prediction of modes in two and threedimensional boundary layers using the method described by Devenport et al. (1999, 

2001) and Devenport and Glegg (2001), previously applied to plane wakes. 

2. Predictions with the code using a constant lengthscale formulation in a fully 

developed turbulence channel flow. Comparison of these boundary layer predictions 

with DNS simulation results of Moser et al. (1999). 

3. Formulation of an improved model using a variable lengthscale proportional to 

mixing length. Turbulent channel flow predictions and comparison with DNS results. 

This work is being carried out in continuous communication and collaboration with the Glegg 

research group at Florida Atlantic University, which will be incorporating mode predictions into 

engine noise calculations. 

INTRODUCTION 

 This project is concerned with the modeling of turbulent boundary layer flows for the 

purpose of predicting trailing-edge noise. Fundamentally, prediction of this noise requires 

information about the two-point vorticity correlations in the boundary layer. In most practical 

circumstances this information cannot be calculated and so noise calculations must depend 

heavily on empirical correlations based on bulk boundary layer properties. 

This project is directed at the application and testing of a new method in which the twopoint velocity and vorticity statistics are predicted explicitly from simple Reynolds stress and 

lengthscale data, such as produced by traditional CFD calculations. This method, initially 

developed by Devenport et al. (1999, 2001) and Devenport and Glegg (2001) for wake flows, is 

being adapted for use in boundary layers and tested against an existing experimental and DNS 

data sets. The results will be incorporated into trailing edge noise predictions, as part of the 

parallel and coordinated study of Glegg (2001). The combined goal of these two studies is the 

development of noise prediction schemes for aircraft engines that are not only more accurate, 

but also allow the key flow noise sources to be identified and thus controlled and reduced. 

William J. Devenport and Nicolas Spitz

AOE Department

Virginia Polytechnic Institute and State University

Blacksburg, Virginia 24061

Phone: 540–231–4456, Fax: 540–231–9632

Email: devenport@aoe.vt.edu

NASA/CR—2003-212099 1
1. Development of a Matlab code for the prediction of two point correlations in two- and threedimensional boundary-layers, based on Devenport et al.s (1999) theory and Devenport and 

Glegg's (2001) improvements. 

2. Initial predictions. Predictions of the two-point correlations for the standard boundary layer 

data sets of Moser et al., 1999, and Adrian et al., 2000 using a constant lengthscale L. 

Comparison of predicts with measurements/DNS results in terms of correlation functions, 

proper orthogonal modes and compact eddy structures. 

3. Transmission of initial results and prediction code to the parallel effort of Glegg (2001). 

4. Refinement of model based upon comparisons with simulation and experimental data, and 

experience in incorporating results in acoustic model. In this task we will focus on how the 

length scale/decay function form can and should be prescribed based upon parameters 

normally available from CFD solutions, and the extent to which the prescriptions improve 

the predictions of the two-point correlations. We will also investigate the operation of the 

method when incomplete turbulence stress profile is available from a CFD solution, such as 

when only turbulence kinetic energy and/or Reynolds shear stress results are available. Final 

comparisons with standard boundary layer data sets, transmission of results to parallel effort 

of Glegg. 

5. Model predictions based on trailing-edge data (e.g. Wang and Moin, 2000). 

6. Analysis, write up and publication of all results. 

The proposal called for completion of items 1 through 3 during the first year of work. While this 

project has only been in place for 61⁄2 months we have nevertheless completed items 1, most of 

items 2 and 3 and some of item 4 during this period. The rest of the report describes this effort 

and the results it has produced so far. 

BACKGROUND OF THE METHOD 

This section describes the basis of the theoretical method we are using to predict twopoint correlations in the turbulent boundary layer. This method was developed for plane wakes 

by Devenport et al. (1999, 2001) and Devenport and Glegg (2001). In this method the two-point 

velocity correlation Rij between fluctuating velocity components ui and uj is expressed as the 

double curl (with respect the two positions r and r') of another function qij (r, r'), ensuring that it 

is divergence free. 

km

jmniklij xx

rrq rrR

∂∂

∂ = ')',( )',( ln

2

ε ε (1) 

The two-point vorticity correlation is then determined simply as the double curl of Rij. For 

homogeneous turbulence it is straightforward to show that 

ij shuq δ ij )()',( 2

2

1 rr −= (2) 

where u is the turbulence intensity, s is the scalar distance between r and r', and h(s) is the first 

moment of the longitudinal correlation coefficient function, i.e. ∫ = s

dssfssh 0 ')'(')( . (See Hinze, 

1973, for the definition of f(s)). For a von Karman turbulence spectrum, 

2

3

1

34 3

4

3

3 3 4 2

/)(/)()()(22)( eee Γ kskKsksh 

  −Γ= (3) 

where ke ≈ 0.75/L and L is the longitudinal integral scale of the turbulence. 

This is a two year project, with the following task list: 

NASA/CR—2003-212099 2
For inhomogeneous turbulence, Devenport et al. proposed an expression analogous to 

equation (3) in which qij is separated into the product of the homogeneous function h(s), that 

controls the manner in which the correlation falls off with separation between points, and an 

scaling function aij(r,r'), i.e. 

 shaq )()',()',( ijij = rrrr (4) 

Substituting equation (4) into (1) we obtain, 















∂∂

∂

+

∂

∂

∂

∂

+

∂

∂

∂

∂

+

∂∂

∂

= ∂∂

∂ =

kmmkkmkm

jmnikl

km

jmniklij

xx

sh

a

x

sh

x

a

x

sh

x

a sh

xx

a

xx

sha R

'

)( )',( ')( )',( )(

'

)',( )( ')',(

'

)()',( )',(

2

ln

lnlnln

2

ln

2

rr

rrrrrr

rr

rr

ε ε

ε ε

 (5) 

where xj and x'j denote, respectively the jth components of r and r'. 

The scaling function aij is chosen so as to make the two-point correlation function consistent 

with the prescribed Reynolds stress field of the flow, i.e. when r=r' (i.e. s=0) the Rij must be 

equal to the Reynolds stress tensor τij. To find a form for aij that will achieve this we first note 

that the value of h(s) and its first derivative tend to zero as s tends to zero, because it is the first 

moment of a correlation coefficient function. Also, its second derivative, 

km xx

sh

∂∂

∂

'

)( 2

tends to -δmk. Thus for s=0, 

 ) ,(),(),()(),( ln rrrrrrrrr δ δ aaaR ooijijmkjmniklijij τ == −ε ε = − (6) 

This equation is satisfied if 

)()(),( 2

1 rrrr a ppijijij = τ − δ τ (7) 

The unexpected simplicity of this relationship is what makes this entire modeling approach 

viable. It shows that it is a trivial matter to prescribe a vector potential correlation function 

consistent with any Reynolds stress field. This result is unexpected because of the otherwise 

second-order differential relationship between the vector potential and velocity correlation 

functions. 

Devenport and Glegg (2001) proposed an expression for aij(r,r') consistent with equation 8 

in terms of the arithmetic average of the Reynolds stress tensor at r and r', specifically 

[ ]

)'()(

)'()'()()()',( 2

1

2

1

2

1

rr

rrrrrr

ijij

ppijijppijijij

bb

a

+=

τ δ τ τ −+−= δ τ (8) 

say. Substituting into equation (6) gives, 













∂∂

∂

++

∂

∂

∂

∂

+

∂

∂

∂

∂ =

∂∂

+∂ =

kmmkkm

jmnikl

km

jmniklij

xx

sh bb

x

sh

x

b

x

sh

x

b

xx

shbb R

'

)( ))'()(( ')( )( )(

'

)'(

'

)())'()(( )',(

2

lnln

lnln

lnln

2

rr

rr

rr

rr

ε ε

ε ε

 (9) 

Apart from being simple, this formulation is attractive because it makes Rij dependent only on 

the derivatives of h(s) and not its actual value. This guarantees decay of the resulting correlation 

at large separations (see Devenport and Glegg, 2001). 

To apply this model to a flow requires as input only the Reynolds stress field τij(r) and the 

single length scale L, required to scale the decay function in equation 3. In effect, an 

NASA/CR—2003-212099 3
inhomogeneous flow is treated as homogeneous turbulence, modulated so as to match the 

Reynolds stress field. However, since the modulation is performed on qln the result is not a trivial 

product of the von Karman spectrum and the turbulence stress field, but a much more complex 

function whose length scales and form vary from point to point in a manner implied by 

continuity and the inhomogeneity, as well as the von Karman form. 

Devenport et al. (1999, 2001) and Devenport and Glegg (2001) tested this model by using it 

to predict the 4-dimensional 2-point correlation tensor of the measured plane wake. They used 

the measured Reynolds stress profile as input, and chose the length scale L to be three-quarters 

of the half-wake width. They found it provided a satisfactory prediction of the the entire twopoint velocity correlation of the wake. The fundamental modes of the wake (i.e. Proper 

Orthogonal Modes and Compact Eddy Structures – see Glegg and Devenport, 2001) that are 

needed for a rigorous aeroacoustic noise calculations (such as leading edge noise produced by a 

blade impacting on the wake) were particularly well reproduced. 

APPLICATION OF THE BASIC METHOD TO A BOUNDARY-LAYER TYPE FLOW 

The first tasks of this project were the development of the Matlab code for performing 

predictions of two-point velocity correlations in boundary-layer type flows, and the application 

and testing of that code through comparison with a standard test case. 

Test Case 

As the standard test case we chose the channel flow DNS simulation of Moser et al. 

(1999), based on the completeness of this data set and initial availability of some explicit 

correlation results. 

Figure 1. Sketch of the channel with the coordinate system and the flow geometry 

Figure 1 shows a sketch of the channel and the coordinate system used for the DNS. In the DNS 

periodic boundary conditions were applied on the x and z directions. The periodic domain was 

chosen by Moser et al. with the intent that the two-point correlation in these streamwise and 

spanwise directions would be essentially zero at half the domain size separation. Simulations 

were available for several Reynolds numbers. 

b 

h=2δ

y,u2 

x,u1 

z,u3 

Flow 

NASA/CR—2003-212099 4
As the test case we 

used the highest Reynolds 

number simulation, 

performed at a Reynolds 

number, based on the 

channel half height δ and 

the friction velocity on its 

walls uτ of 590. The 

statistics extracted from the 

simulation are available at 

http://www.tam.uiuc.edu/Fa

culty/Moser/channel. These 

data include the two-point 

correlations in the 

streamwise and spanwise 

directions for fixed y

positions. Correlations in 

the y-direction, while 

implicit in the Moser data 

set are not explicitly 

available. 

Application of the 

current 2-point correlation prediction method requires the specification of the Reynolds stress 

field (for equation 8) and the single constant lengthscale used in equation 3. The continuous 

Reynolds stress distributions required by the method were obtained by interpolating the DNS 

results, which were available for 129 y positions across the half height of the channel, figure 2. 

The length scale for the model was chosen by a trial and error method aiming at fitting the DNScomputed time delay correlation 

coefficient function for u2

fluctuations at the center of the 

channel. Figure 3 shows the fit 

obtained with the final value of 

the lengthscale L=0.28δ, used 

for the model predictions. 

Results 

Figures 4 and 5 compare 

two-point spatial correlation 

functions predicted using the 

current constant lengthscale 

model, with those given by the 

DNS data set. The pictures 

compare the correlations as a 

function of streamwise 

separation for points at the same 

DNS 

Figure 3. Comparison of the decay of the correlation 

coefficient of u2 fluctuations with streamwise separation 

at the channel centerline. The horizontal axis 

represents streamwise separation normalized on δ. 

Figure 2. Reynolds stress profiles from the Moser data set used as input to 

the two-point correlation predictions. Horizontal axis is y/δ. Symbol u11 is 

used to indicate the Reynolds stress in the u1 direction, for example. 

NASA/CR—2003-212099 5
Figure 4. Comparison with streamwise correlation functions predicted using the model and 

given by the DNS solution (denoted as ‘experimental values in the legend). Stated y

locations and x-positions on horizontal axes are normalized on the channel half width. 

NASA/CR—2003-212099 6
Figure 5. Comparison with streamwise correlation functions predicted using the model and 

given by the DNS solution (denoted as ‘experimental values in the legend). Stated y

locations and z-positions on horizontal axes are normalized on the channel half width. 

NASA/CR—2003-212099 7
z and y location, and correlations as a function of spanwise separation for points at the same x

and y (these are all the types of correlations available in the reduced DNS data set). These 

correlation function comparisons are presented for four different y locations. Note that these 

figures use the notation u,v and w to denote the velocity components u1, u2 and u3 and thus, for 

example, the correlation R12 is denoted by the symbol Ruv. 

At the center of the channel (y=1) the correlation functions predicted with the model 

generally compare well with those from the DNS. Note that the large deviations in the values of 

Ruv at this station from the DNS are merely the result of statistical uncertainty. The value of this 

correlation must be zero at the channel center, by symmetry, as predicted by the model. As y 

decreases the prediction between DNS and model worsens. Some of the disagreement is 

accounted for by the fact that correlation functions given by the DNS do not always decay to 

zero (as the computers intended). This lack of realism in the DNS data is most apparent in the 

streamwise decay of Ruu at y=0.25δ (figure 4). However, ignoring this it is apparent that, closer 

to the wall, the model begins to substantially under predict the rate of decay of most of the 

correlations. This problem is most likely associated with the fact that we are attempting to use a 

single, constant, lengthscale to represent the entire flow, whereas in reality the turbulence 

lengthscale in the channel flow decreases rapidly as the wall is approached. 

 Figure 6 provides a 

somewhat different view of the 

correlation function predicted 

using the model. Plotted are the 

9 components of the correlation 

as functions of the y-coordinates 

of the two points, for zero 

separation in z and x. While no 

reduced DNS data are available 

for comparison with these 

results, they are quite revealing 

of the problems associated with 

the constant lengthscale 

assumption. 

These maps show, as 

expected strong correlations 

along the diagonals (y=y') where 

they represent the Reynolds 

stress profiles. Furthermore they 

show the modeled correlation 

functions to be symmetric about this line, as they must be. However, the Ruu and Rww correlation 

functions show very unrealistic behavior close to the y and y’ axes, indicating that velocity 

fluctuations near the wall are closely correlated with fluctuations across the entire channel, 

where as those near the channel center are not. This unlikely behavior turns out to be a 

consequence of the constant lengthscale assumption, as is demonstrated below. 

 Additional to the calculations and analysis presented here, the constant lengthscale 

calculations have also been used to predict the one-dimensional proper orthogonal modes of the 

y/δ

y'/δ

Figure 6. Two-point correlation maps for zero x and z

separation as a function of y locations of the two points. 

Ruw 

Ruv 

Ruu 

Rvw 

Rvv 

Rvu 

Rww 

Rwv 

Rwu 

NASA/CR—2003-212099 8
channel flow. The grid dependence of the model solution has also been investigated. For details 

see Spitz (2002). 

EXTENSION OF THE METHOD TO ALLOW PRESCRIPTION OF A VARIABLE 

LENGTHSCALE 

The above results that allowing a variable lengthscale to be prescribed in the model could 

improve predictions. Prescribing such a lengthscale is still consistent with the ultimate 

application of the model, i.e. as a post processor for CFD results that provides 2-point correlation 

estimates, since almost all standard turbulence models used in CFD codes give as output some 

lengthscale as a function of position throughout the flow. 

Devenport and Glegg (2001) and Devenport et al. (2001) discussed how a variable 

lengthscale and anisotropic decay function could be accommodated within the model framework. 

They argue that such variations of this type can be incorporated into the model by replacing 

equation (9) with, 

km

ss

jmniklij xx

bshbsh R

∂∂

+∂ = '))'()',,()(),,(( )',( lnln

2 rrerre

rr ε ε (10) 

where the dependence of h() on es, the vector direction of separation, allows for the prescription 

of an anisotropic decay function, and the dependence on r allows for point to point changes in 

the prescription of the decay function form, or its lengthscale. If we are only concerned with 

varying the lengthscale we can write, 

km

jmniklij xx

bLshbLsh R

∂∂

+∂ = '))'())'(,()())(,(( )',( lnln

2 rrrr

rr ε ε (11) 

with h() still given by equation (3). As discussed by Devenport and Glegg (2001) this type of 

manipulation of the decay function does not alter the simple relationship between the scaling 

function b() and the prescribed Reynolds stress field that is the basis of the method. 

As a first step we chose to 

prescribe L as proportional to a 

standard mixing length distribution. 

The van Driest (1956) mixing length 

formulation prescribes a value that 

increases from zero at the wall until it 

reaches a value of 0.09δ. From this 

point on the mixing length is 

prescribed as constant at 0.09δ. The 

variation in the near-wall region is 

given by the equation. 

 )] /exp(1[ +++ −−= Ayy l m δ νκ

 (12) 

where δ ν is the viscous lengthscale, 

κ = 0.41 and A+ the van Driest 

constant equal to 26 For Moser et al.’s 

(1999) channel flow this reaches a 

value of 0.09δ at y/δ=0.225. The 

constant of proportionality between L

y/δ

y'/δ

Figure 7. Two-point correlation maps for zero x and z

separation as a function of y locations of the two points, 

Ruw 

Ruv 

Ruu 

Rvw 

Rvv 

Rvu 

Rww 

Rwv 

Rwu 

lengthscale proportional to mixing length. 

NASA/CR—2003-212099 9
and lm was chosen so that the value of L at the channel center would be the same as in the 

constant lengthscale calculations. Specifically, we set L=0.0053lm. 

 Figure 7 shows the same y- correlation data plotted in figure 6, recomputed with the 

lengthscale varying according to this prescription. The introduction of the variable lengthscale 

appears to eliminate the sudden increase in the extent of the Ruu and Rww correlations close to the 

walls. Comparisons similar to those shown in figures 4 and 5 (see Spitz, 2002) show improved 

(although still imperfect) agreement between the variable lengthscale model and the DNS. 

PLAN FOR FY03 

Our sense of the above results, and or our progress so far, is that they provide an initial 

demonstration of the viability of predicting the full velocity correlation function, and thus the 

velocity and vorticity modes of a turbulent wall layer using the method of Devenport et al.

(1999, 2001) and Devenport and Glegg (2001) modified so as to allow for the prescription of a 

variable lengthscale. Indeed, we are currently working on an additional variable lengthscale 

formulation, based on the turbulent macroscale k3/2/ε which is available from many widely used 

turbulence models, such as the k-ε model. However, it is also apparent that the reduced data 

available from Moser et al.’s (1999) simulation is limited when it comes to comparing with the 

model results. The reduced data includes no y-correlations and thus cannot be compared with 

pictures such as figure 6 and 7. Without y-correlations the DNS data cannot be used to determine 

the all important Proper Orthogonal or Compact Eddy modes used in noise calculations, and thus 

no comparisons of these are possible either. 

To remedy this situation we have managed to obtain access to the raw (instantaneous) 

DNS results, and have begun reducing them ourselves in order to obtain the needed y-correlation 

information. We will also be looking at other data sets (which are also incomplete, but in other 

ways) for comparative data, in particular the boundary layer measurements of Adrian et al., 

(2000). 

Once we are able to validate the modal predictions of the model, we will provide the first 

version of this model to the Glegg group, and begin calculations of the Wang and Moin (2000) 

trailing edge flow. We therefore expect to be close to the completion of the project at the end of 

FY03. 

REFERENCES 

Devenport W J, Muthanna C, Ma R and Glegg S A L, "Measurement and Modeling of the Twopoint Correlation Tensor in a Plane Wake for Broadband Noise Prediction", First 

International Symposium on Turbulence and Shear Flow Phenomena, Santa Barbara, CA, 

September 12-15, 1999. 

W J Devenport, C Muthanna and S A L Glegg, 2001, "Two-Point Descriptions of Wake 

Turbulence with Application to Noise Prediction", AIAA Journal, vol. 39, no. 12, pp. 

2302-2307. 

Devenport W J and Glegg S A L, 2001, "Modeling the two-point space time correlation of 

turbulence in a fan wake type flow", 7th AIAA CEAS Aeroacoustics Conference, 28-30th

May 2000, The Netherlands. 

Glegg S A L, 2001, “Noise Generated by Fans with Supersonic Tip Speeds”, Parallel grant under 

NRA-01-GRC-02 Aerospace Propulsion and Power Base Research and Technology 

Program. 

Hinze J O, 1973, Turbulence, McGraw Hill, New York. 

NASA/CR—2003-212099 10
Moser R D, Kim J, and Mansour N N, 1999, " Direct numerical simulation of turbulent channel 

flow up to Re-tau=590", Physics of Fluids, vol. 11, pp. 943-945. 

Spitz N, 2002, “Prediction Of 2-Point Statistics In A Turbulent Boundary Layer”, AOE Dept., 

Virginia Tech, June. 

Wang M and Moin P, 2000, "Computation of trailing-edge flow and noise using large-eddy 

simulation", AIAA Journal, vol 38, pp. 2201-2209. 

NASA/CR—2003-212099 11
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E–13752

WU–708–87–23–00

NAG3–2710

17

Predicting Modes of the Unsteady Vorticity Field Near the Trailing Edge

of a Blade

William J. Devenport and Nicolas Spitz

Vorticity; Aerodynamic noise; Trailing edges

Unclassified -Unlimited

Subject Category: 02 Distribution: Nonstandard

Virginia Polytechnic Institute and State University

AEO Department

215 Randolph Hall

Blacksburg, Virginia 24061

Project Manager, Edmane Envia, Structures and Acoustics Division, NASA Glenn Research Center, organization

code 5940, 216–433–8956.

Progress on predicting modes of the unsteady velocity/vorticity field of a turbulent boundary layer from Reynolds stress

statistics is described. Prediction of these modes, that provide the source terms for trailing edge noise predictions in

aircraft engine fans and other configurations, will allow for the first time detailed viscous flow effects to be included in

such noise calculations. The key accomplishments of this work in FY02 are 1. The development of a Matlab code for

the prediction of modes in two- and three-dimensional boundary layers, previously applied to plane wakes; 2. Predictions with the code using a constant lengthscale formulation in a fully developed turbulence channel flow. Comparison

of these boundary layer predictions with available DNS simulation results; and 3. Formulation of an improved model

using a variable lengthscale proportional to mixing length. Turbulent channel flow predictions and comparison with

DNS results. This work is being carried out in continuous communication and collaboration with the Glegg research

group at Florida Atlantic University, which will be incorporating mode predictions into engine noise calculations.