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Determination Of Influence Boundaries In CFD From Acoustic Intensity Distribution

H.Q. Yang 1 CFD Research Corp., 215 Wynn Dr. Huntsville,, AL, 35805

Abstract

Using influence region technique in CFD leads to simpler grid generation, faster computing time, and the unique capability of utilizing multiple equation sets or multiple flow solvers. We have successfully derived, implemented, and demonstrated the feasibility of using acoustic intensity distribution as the guideline in selecting influence boundaries. This approach is first principle–based, and can be solved in time and frequency domain or in steady state. The formulation explicitly depends on Mach number, angle of attack and body geometry. By comparing the results between full geometry and reduced-geometry computations, it is demonstrated that the present approach can estimate errors a priori due to: far field boundary selection, local geometrical change, different solver choice/selection, and local grid refinement.

I. Introduction A. Background

With computational fluid dynamics (CFD) becoming more accepted and more widely used as a design and analysis tool, there is an increasing demand for improved solution accuracy and improved efficiency. Using influence region technique in CFD has the advantage of simpler grid generation, faster computing time, and the facility of multiple equation sets or multiple flow solvers.

The so-called influence region is an intermediate sized domain, where multiple equation sets or multiple flow solvers can be used, and grid can be arranged to be either structured or unstructured. The approach has the following advantages:

a) Different Solver for each influence region:. Due to complex physics involved in the external flow, different solvers can be applied to different regions of the flow. For example, Miller et al. [1] have developed a parabolized Navier-Stokes (PNS) algorithm which automatically detects and measures the extent of the embedded (circulation) regions (separated by influence boundary) that produce upstream effects. With the influence boundary determined from a correlation function, the standard space-marching PNS mode has been used in the supersonic flow, and iterative PNS has been used for the embedded region. The calculation has shown excellent agreement with full NS solver and demonstrated significant reduction in computer time and storage. Another example is the use of Euler solver and Full N-S solver combination for flow around bluff bodies. Euler solver can be applied at distances far away from the body(far-field) and N-S code can be used to resolve the boundary layer and separation effect.

b) Different grid system for each influence region: A combination of structured and unstructured grids can be used in different regions of the flow. Very high orders of accuracy can be achieved by using structured grid in certain region, while several geometric and solution adaptive strategies can be used to create unstructured grid locally to resolve flow features.

c) Simplified grid generation. Due to the flexibility in selecting influence boundary, the computational domain can be simplified to a shape and orientation that is easier to generate. Davis et al. [2] have developed a minimized domain CFD method for store separation. They have limited the computational domain for the store to a relatively small area immediately surrounding the store thereby excluding the complex geometric features of the entire aircraft/store system. The influence boundary condition has been

1 Chief Scientist, Research, CFD Research Corp., 215 Wynn Dr. Huntsville,, AL, 35805, Senior AIAA Member

46th AIAA Aerospace Sciences Meeting and Exhibit7 - 10 January 2008, Reno, Nevada

AIAA 2008-681

Copyright © 2008 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

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developed such that the flowfield data from an aircraft flowfield solution is applied as interface boundary conditions. They have found that this technique can calculate store flowfields and loads in a fraction of the time of a full CFD simulation.

As shown in the previous studies, influence zone method offers many significant benefits for CFD. The use of influence region to sub-divide a computational domain can also minimize the inter-dependency between spatial discretizations and physical models [3]. This contributes to faster development of new numerical schemes by avoiding their impact on models already implemented.

The use of influence region is similar to overset method, where the solution process resolves the geometrical complexity of the problem domain by using separately generated but overlapping structured discretization grids that periodically exchange information through interpolation(see Figure 1). Overset method will also benefit from the advancement of influence region technique in that this method will provide a guide to construct overset grids that minimize regions where grid-to-grid interpolation may affect computed solutions.

B. Present Approach

A influence region technology has been developed for CFD by using a first principle based method for the determination of influence boundaries. The present method uses numerical solution to an acoustic equation using acoustic intensity factor as a monitoring quantity. In our approach, given the free stream Mach number, angle of attack, and model geometry, a CFD solution based on coarse mesh and simple physical model is computed along with a quick solution of acoustic wave equation based on Ffowcs-Williams-Hawkings formulation (FW-H equation). The acoustic solution, expressed in terms of the decay of the acoustic signal or numerical noise (due to grid refinement or different physical model), is used to determine the distribution of the acoustic intensity. Isotropic value of the intensity is used to quantify the influence regions.

C. Objectives The objective of this research is to develop an efficient, first principle based algorithm to locate influence

boundaries for specified configuration and their flow field simulation. The specific objectives are: 1. Implement a numerical solution to FW-H equation in an existing unstructured CFD-solver; 2. Evaluate the acoustic characteristics using the second order spatial and temporal schemes (central in space

and Newmark scheme in time) to the FW-H equation; 3. Demonstrate the feasibility of determining influence boundaries in external inviscid flows given the

freestream Mach number, angle of attack, and model geometry; 4. Explore the feasibility of solving CFD equations with high accuracy and high efficiency, despite the lack of

detailed knowledge of the process prior to the calculation.

II. Theoretical Background D. Derivation of Acoustic Intensity Equation from FW-H Equation

The theoretical basis for the analysis of sound generated by a body moving in a fluid is represented by the FW-H equation [4], which can be derived from the basic conservation laws of mass and momentum written in terms of generalized functions. This formulation allows jumps in flow quantities across the surface of the moving body. By neglecting viscosity effects (the fluid-body dynamic interaction is described by the scalar pressure field on the body surface) and by assuming that the fluid is compressible and undergoes transformations with negligible entropy changes the FW-H equation reads (see detail derivation from the work of Brentner and Farassat [5]):

Figure 1. Example of use of overset grid for CFD application

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( )[ ] ( )[ ] ( )[ ] ( )[ ]fBt

fLx

TfHxx

fHxx

ct i

iij

jiiio δδρ

∂∂

+∂∂

−∂∂∂

=′⎥⎦

⎤⎢⎣

⎡∂∂∂

−∂∂ 22

22

2

(1)

where co is the ambient sound speed, and

,2ijoijjiij cPuuT δρρ ′−+= ( ),vuun̂PL nnijiji −+= ρ ( )vnno uuvB −+= ρρ (2)

In the above equations, f(x,t) defines a closed non-deformable moving surface. f < 0 in the volume interior to the surface, and f > 0 in the volume exterior to the surface. The Lighthill stress tensor Tij, contributes to the quadrapole term on the right-hand side of Eq. (1), while Li and B contribute, respectively to the dipole and monopole terms. H(f) and δ(.) respectively denote the Heaviside step function and the Dirac delta function. The inviscid compressive stress tensor ijij pP δ= is used here where p denotes the unsteady part of the pressure, and δij is the

Kronecker delta. The quantities un and vn are respectively the components of the fluid and surface velocities that are normal to the surface f = 0, i.e., iin n̂uu = and iin n̂vv = . The summation convention applies to repeated indices. It is re-emphasized here that Eqs. 1 and 2 are valid whether the surface is on or off the solid body. When the surface f is coincident with the solid body, the expressions for Li and B are simplified by setting nn vu = in Eq. 2.

E. Simplification of FW-H Equation In the present analysis, we assume that 1. f(x,t) is the solid surface of the object we are to study, so that nn vu = . 2. the motion of the solid surface is at constant speed, so that the last term on the right hand side of Equation

(1) is zero. It should be noted the original Lighthill analogy is written for an observation region at rest [5]. For the case of a

uniform flow, Ffowcs Williams and Hawkings proposed the use of a Lagrangian co-ordinate transformation assuming the surface is moving in a quiescent fluid. As will be shown in a later section, this study considers the homogenously convected wave equation in the fluid region (f>0) only. This consideration simplifies the FW-H equation into the original Lighthill equation:

[ ]ijji

2

ii

22o2

2

Txx

'pxx

ct ∂∂

∂=⎥

⎦

⎤⎢⎣

⎡∂∂∂

−∂∂

(3)

Convected FW-H Equation The wave equation (3) is derived using a coordinate system that is at rest with respect to the fluid. However,

the co-ordinate transformation for a frame of reference moving with a speed Ui in xi direction is:

,'tU'xx iii += 'tUx'x iii −= , 'tt = (4)

This modifies the time derivative term in Equation (3) as follows:

't'p

'x'pU

't'p

't'x

'x'p

t'p

ii

i

i ∂∂

+∂∂

=∂∂

+∂∂

∂∂

=∂∂

(5)

and

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

∂∂

+⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

∂∂

=⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

∂∂

=∂∂

't'p

'x'pU

't't'p

'x'pU

'xU

't'p

x'pU

tt'p

ii

ii

jj

ii2

2

2

2

i

2

iji

2

ji 't'p

'x't'pU2

'x'x'pUU

∂∂

+∂∂

∂+

∂∂∂

= (6)

With this transformation, the Lighthill stress tensor is now:

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,cP)Uu)(Uu('T ij2oijjjiiij δρ′−+−−ρ= (7)

and

'x'x

'pxx'p

ji

2

ji

2

∂∂∂

=∂∂

∂ (8)

The acoustic equation can be rewritten as:

'T'x'x'x'x

'pc'x't

'pU2'x'x

'pUU't

'pij

ji

2

ji

22

oi

2

iji

2

ji2

2

∂∂∂

=∂∂

∂−

∂∂∂

+∂∂

∂+

∂∂

(9)

By excluding the “prime” and defining Mi=Ui/co, the above equation can be expressed as:

ji

ij

jiji

iio xx

TxxpMM

xtpMc

tp

∂∂

∂=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

∂∂∂

−+∂∂

∂+

∂∂ 222

22

2

)1(2 (10)

Equation (10) can be solved with or without the quadrupole term depending on the desired physics. Without the quadrupole term, the equation can easily be solved without detailed knowledge of the flow around the object, so much so even analytical solution can be obtained. If flow separation is important such as in the wake region, one needs to include the quadrupole term. In such a case, the Lighthill stress tensor Tij in quadrupole term has to be computed from a CFD solution.

Equation (10) can be solved by integrating over a control volume, and discretizing the derivative terms using central differencing scheme. Newmark scheme is employed for the second derivatives in time. This choice of schemes gives second order accuracy in both space and time.

The above equation is very similar to electro-magnetic equation in that it can be solved in time domain, as well as in frequency domain. By using the following substitution for p:

tipep ω= (11)

Equation (10) transforms to

0)1(22

22 =⎥⎥⎦

⎤

⎢⎢⎣

⎡

∂∂∂

−+∂∂

+−ji

jii

io xxpMM

xpiMcp ωω (12)

By separating p into real and imaginary parts:

ippp ir += (13)

Equation (12) is split into the following two equations:

0)1(22

22 =⎥⎥⎦

⎤

⎢⎢⎣

⎡

∂∂∂

−+∂∂

−+−ji

rji

i

iior xx

pMMxpMcp ωω (14)

0)1(22

22 =⎥⎥⎦

⎤

⎢⎢⎣

⎡

∂∂∂

−+∂∂

+−ji

iji

i

rioi xx

pMMxpMcp ωω (15)

For a steady state scenario, the equation simplifies to:

0)1(2

=∂∂

−ji

ji xxpMM (16)

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As can be seen, this is a Poisson equation, with pressure fluctuation depending on Much number Mi, Mj, and angle of attack, which is defined as:

x

y

MM

tan =α (17)

At zero angle of attack,:

0)1( 2

2

2

2

2

22 =

∂∂

+∂∂

+∂∂

−zp

yp

xpM x (18)

The above equations can be solved by finite difference method, finite volume, boundary element, or panel method. The panel method has the advantage that no grid generation is necessary.

For the purpose of proving the concept, an existing CFD solver has been modified to solve equation (16).

F. Numerical Solution of Acoustic Intensity Equation A control-volume based solver has been implemented to solve equation (16). The solution domain is

subdivided into a finite number of contiguous control volumes in arbitrary combination of: 1. triangles, quadrilaterals or polylaterals in 2D; and 2. tetrahedrals, hexahedrals, prisms, pyramids, and polyhedra in 3D. The control volumes are defined by coordinates of their vertices, which are assured to be connected by straight

lines. All dependent variables are stored at the centroid of the control volume while fluxes are evaluated at the cell faces.

III. Application Of Acoustic Intensity Method G. Initial Test with Mach = 0.0 for 2D Airfoil with a Flap

One of the simplifications to equation (16) is when Mach number is set to zero. This will reduce the original equation to a Laplace equation. Figure 2 depicts the configuration and geometry for the initial test case. A coarse unstructured grid was generated around the above airfoil/flap system, as shown in Figure 3.

Figure 2. Proposed 2D airfoil/flap configuration

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The solution of Equation (16) with a boundary condition of p=1 on the airfoil surface, and p=0 on the outside is

carried out. The ideas behind the above solution and boundary conditions are: 1. Lift and pitch moment are strong function of pressure on the airfoil and flap. An error in the estimation of

pressure on the airfoil and flap surfaces will lead to an error in the lift and pitch moment. 2. If we select our “full” size computation domain as 40 cord length from the center of the airfoil, an error

associated with distribution of p reflects the error due to the change of the computational domain (from 40 c to 20c, to 10c, and to 5c)

3. From the solution of acoustic intensity factor, we can determine the error of influence boundary. To further explain the above point, the distribution of p is shown in Figure 4 below.

As one can see at Mach number=0, the acoustic intensity factor has a circular shape, which is consistent with

results of incompressible flow case where an acoustic signal will propagate in all directions. By reducing the domain size by 20c in each direction for this configuration, a 1% error can be expected from the computations by

Figure 4. Distribution of acoustic intensity factor around an airfoil with a flap for Mach =0.0

Figure 3. A coarse unstructured mesh around airfoil and flap. The boundary condition for

pressure fluctuation is also specified

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imposing constant inflow conditions of p=p0, u=u0, and v=v0 instead of pb(x,y),ub(x,y), and vb(x,y) solution from 40c. The constant contour of log p gives the indication of the error for this flow solution.

IV. Validation Study For Airfoil With A Flap H. Errors Estimation Associated with the Selection of Far Field

One of the important issues the present acoustic intensity factor method can address is the selection of far field boundary conditions. Mathematically, the out boundaries need to be selected far away so that the specified boundary values will have minimum effect on the solution itself. An airfoil with flap as shown in the above section will be used as a test case for this study. The configuration and geometry and the results are given in Figure 5.

A coarse unstructured grid was generated around the above airfoil/flap system, as shown in Figure 5. During

this study a coarse mesh is used to solve for the acoustic intensity factor to prove the concept. For the condition of Mach number 0.25, 0° of angle of attack, and 0° degree flap angle, the distribution of

acoustic intensity factor is also shown in Figure 5. The p distribution and errors indicate that a far-field boundary condition of ∞∞∞ === pp,vv,uu applied to the out-boundary of a computational domain with 20 chord length will result in a 1% error when compared to the full domain solution(defined as 40 chord length away from the airfoil). Similarly, a 5% error is expected for 10-chord length, and 10% error is expected when using a 4-chord length side boundary when using the far field boundary.

To validate the above prediction, full Navier-Stock calculations have been performed for the above conditions. The airfoil/flap configuration has been modeled with four different domain sizes with the same internal grid to eliminate grid variation effect (see Figure 6). A total of 64 cases have been setup with two different angles of attack (AOA), four different flap angles, and two different freestream Mach numbers. The specific conditions considered are:

1. 0 and 5 degree AOA 2. 0, 10, 20, and 30 degree Flap angle 3. 4, 10, 20, and 40 Chord domain size 4. 0.25 and 0.8 freestream Mach Number The cell count for the various grids varied between 29K to 33K. Air is treated as an ideal gas with constant

thermal properties. A standard k-epsilon turbulence model is applied to capture the turbulence effects. Fixed velocity condition and constant pressure condition are imposed at the inlet and outlet boundaries respectively. The domain sizes and the various flap configurations are shown in Figure 6.

Figure 5. Left: Airfoil and flap configuration and set up for acoustic intensity solution; Right: the

acoustic intensity distribution and the associated error

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Figure 7 shows the comparison of two typical cases of present acoustic intensity prediction of error and the

actual error from the CFD simulation. The error associated with drag, lift and moment are compared. The trend of error variation with the distance from airfoil surface(chord length) and also the order of magnitude of the error prediction agree well with actual CFD. There are total 16 such plots for different scenarios (varying AOA,Mach Number and Flap Angle) considered that show similar results.

I. Errors Estimation Associated with the Local Geometrical Change One of the important applications of influence boundary method is to estimate the error associated the local

geometrical variation. Figure 8 shows a situation with the desire to change the angle of the flap, while keeping the input value at the computational boundary the same.

Figure 6. Computational grid for validation study using full Navier-Stokes equation

Figure 7. Comparison of present prediction with the actual CFD calculation

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Figure 9(a) shows how the boundary condition should be specified for the solution of the acoustic intensity

factor. The error distribution when the flap angle changes from 0° to 20° is the superposition of two solutions: 1) With flap angle at zero degree, and a p’=–dp0_flap, and dp/dn=0 at the airfoil surface; and 2) With flap angle at 20 deg., p’=1, and dp/dn=0 at the airfoil surface.

dp0_flap is the ratio of average pressure on the flap at 0° flap and that of 20° flap from a potential solution. The resulting solution using the Figure 9(a) boundary conditions is shown in Figure 9(b). This solution is compared to

Figure 8. Application of influence boundary: local boundary values are fixed, and internal geometry is allowed to change. The present acoustic method can estimate the associated error

(a) Boundary Condition for the Acoustic Solution

(b) Acoustic Intensity Distribution for the above Boundary Condition

(c) Comparison of Present Prediction with Actual CFD Simulation Figure 9. Error estimate for the flap angle change from the present acoustic method

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actual CFD calculation and the error distributions are tabulated in Figure 9(c). The comparison shows a good match in the order of the error magnitude. This study shows that using 4-chord length as influence boundary one can expect a maximum of 3% error.

J. Errors Estimation Associated with the Use of Different Solver CFD technology has evolved from early potential method, to Euler equation solver, to Navier-Stokes equation

solver, and to Large Eddy Simulation. With this evolution, the complexity and computational time have also increased significantly. Ideally, one would like to use the different solvers for different regions of the flow so that the computational cost can be reduced. For example, for an external aerodynamic problem, the flow experiences very small disturbance away from the aircraft/airfoil/wing bodies. In the far field, a potential solution will be accurate enough. However, near the aircraft/wing body, the local turbulence flow and flow separation requires the full Navier Stokes solution. At the trailing eddy, to resolve vortex shedding, a LES model will be a better choice. If one were to adopt a combination of these solvers, the issues of solver-boundary selection and potential error estimation arise. The present acoustic method is able to address these issues.

Figure 10 illustrates the procedure. Initially, a panel method without grid or an Euler method with coarse

method is used to solve the far field of the flow. The boundary condition is specified at an interface 4chord length away (considered as influence boundary). With the prescribed boundary value of u, v and pressure obtained from less complex solver (either full potential or Euler equation), a Navier-Stokes solver is applied. Since the boundary velocity and pressure are from the potential method or Euler method, they are not the true values if one were to use the N-S solver for the whole domain. To estimate these errors involved, an acoustic intensity distribution is solved for with p=1 at the airfoil and flap body, and p=0 at 40 chord length boundary (which is taken as full domain). The analysis as shown in Figure 10 results in a relative error of 10%.

When the initial boundary value at the influence boundary is taken from an Euler solution, and then a N-S

solver is used at the 4-chord length region, there will be a change in the solution of: pN-S-pEuler LN-S-LEuler MN-S-MEuler

Since the boundary values of u, v and pressure are fixed, this will cause errors of 10% *( p N-S-p Euler) in pressure coefficient 10%* (LN-S-LEuler) in lift coefficient 10%*( MN-S-MEuler) in moment coefficient

Figure 11 shows the comparison of present prediction and the actual error. Again one can see that errors are well predicted from the present method.

Figure 10. Use of different solver and the associated error estimate using the present method

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K. Errors Estimation Associated with the Local Grid Refinement One of the advantages of using influence boundary is the freedom of local refinement. This is shown in Figure

12, where one can use pb(x,y), ub(x,y) and vb(x,y) as boundary condition, and locally refine the grid. Question is what will be the error from the grid refinement caused by using the same BC value from the coarse mesh ? If we define CLc and CMc as the lift and moment coefficients from the coarse mesh, and CLf and CMf as those of fine mesh, we will expect that when grid is refined, the solution will change. But since the boundary values of u, v, and p are fixed, it will cause an error of δ%, in reference to Figure 12 from 40c to 20c:

δ% (pf-pc) in surface pressure δ% (CLf-CLc) in lift coefficient δ% (CMf-CMc) in moment coefficient Where δ% is the error calculated from the present acoustic intensity factor on the 20c (or 10c, 5c) boundary.

We observe that: 1. When a large influence boundary is chosen, δ% error is relatively small, so that the net error will also be

small. 2. When a small influence boundary is chosen, δ% error could be high, so that an improvement in lift and

moment coefficients may be over taken by the error associated with the unrealistic boundary values at influence boundary: pb(x,y),ub(x,y),vb(x,y). Therefore, it is important that when using a small influence boundary, the boundary values are updated.

V. Summary This study has successfully demonstrated that the acoustic intensity method has many unique merits for the

determination of influence boundary. Some of the salient features of this method are summarized below: 1. It is first principle based approach with strong built in physics. 2. It can be solved without generating any volume grid, or it can be solved very quickly on a very coarse

volume mesh 3. It can be solved in time domain or in frequency domain; 4. The formulation depends explicitly on Mach number and angle of attack;

Figure 11. Error comparison of the present acoustic method and actual error for the use of different solver

Figure 12 Error estimate for local grid refinement

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5. It can be applied to multi-time scale and multi-length scale problems; 6. It can be easily incorporated into a grid generation package; 7. It has been validated for the error estimate on the far field boundary condition selection; 8. It has been validated for the error estimate on the local geometrical change; 9. It has been validated for the error estimate on the use of different solvers; 10. It is capable of the error estimate on the local grid refinement.

Acknowledgments

The present effort was supported through a SBIR Program under contract number No: FA8650-05-M-3529 funded by AFRL, Wright Patterson AFB. The technical input provided by Dr. Gregory Brooks, the contract monitor, is greatly acknowledged.

References 1J. H. Miller, J. C. Tannehill, and S. L. Lawrence, “Parabolized Navier-Stokes Algorithm for Solving

Supersonic Flows with Upstream Influence”, AIAA Journal, Vol. 38, pp.1837, 2000. 2M. B. Davis, T. J. Welterlen, K. J. Corfeld, “Minimized Domain CFD for Store Separation”, AIAA-2003-1245,

2003. 3Lange, C.F.; Durst, F.; and Breuer, M., “Momentum and Heat Transfer from Cylinders in Laminar Crossflow

at 10-4 < Re < 200,” International Journal of Heat and Mass Transfer, No. 41, pp. 3409-3430, 1998. 4Fowcs Williams, J. E., and Hawkings, D. L., “Sound Generation by Turbulence and Surfaces in Arbitrary

Motion,” Philosophical Transactions of the Royal Society of London, Vol. 264A, May 1969, pp. 321–342. 5K. S. Brentner and F. Farassat, “An Analytical Comparison of the Acoustic Analogy and Kirchhoff

Formulation for Moving Surfaces”, AIAA J, Vol. 36, No. 8, pp. 1379, 1998.