NASA/TM--2001-210703 FEDSM-2001-18136

GAP Noise Computation by the

CE/SE Method

Ching Y. Loh

Taitech, Inc., Cleveland, Ohio

Sin-Chung Chang

Glenn Research Center, Cleveland, Ohio

Xiao Y. Wang

Taitech, Inc., Cleveland, Ohio

Philip C.E. Jorgenson

Glenn Research Center, Cleveland, Ohio

Prepared for the

2001 Fluid Engineering Division Summer Meeting

sponsored by the American Society of Mechanical Engineers

New Orleans, Louisiana, May 29-June 1, 2001

National Aeronautics and

Space Administration

Glenn Research Center

February 2001

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GAP NOISE COMPUTATION BY THE CE/SE METHOD

Ching Y. Loh

Taitech, Inc.

Cleveland, Ohio 44135

Sin-Chung Chang

National Aeronautics and Space Administration

Glenn Research Center

Cleveland, Ohio 44135

Xiao Y. Wang

Taitech, Inc.

Cleveland, Ohio 44135

Philip C. E. Jorgenson

National Aeronautics and Space Administration

Glenn Research Center

Cleveland, Ohio 44135

Abstract

A typical gap noise problem is considered in

this paper using the new space-time conservation element and solution elemem (CE/SE)

method. Implementation of the computation is

straightforward. No turbulence model, LES

(large eddy simulation) or a preset boundary

layer profile is used, yet the computed frequency agrees well with the experimental one.

1 Introduction

Gap noise is an important issue in aerospace and auto

industries. For example, modern automobiles are required to be comfortable and quiet, however, due to vortex shedding in high speed air flow over door gaps, unpleasant tonal noise may be generated. Strong oscillations occur in a feedback cycle in which the vortices

shed from the upstream edge of the gap convect downstream and impinge on the other edge, generating acoustic waves that propagate upstream to excite new vortices.

Numerical simulation of such a complicated process requires a scheme that can : (a) resolve acoustic waves

with low dispersion and numerical dissipation, (b) handle nonlinear and discontinuous waves (e.g. shocks), and

(c) have an effective (near field) non-reflecting boundary

condition (NRBC). The new space time conservation element and solution element method, or CE/SE for short,

is a numerical method that meets the above requirements

[ 1-4]. A detailed description of the 2-D CE/SE Euler

scheme can be found in [1,2], only a brief sketch is given

here.

In nature, the new method may be categorized as a

finite volume method, where the conservation element

(CE) is equivalent to a finite control volume (or cell)

and the solution element (SE) can be understood as the

cell interface. However, due to its rigorous treatment of

the fluxes and geometry, it is different from the existing

schemes. As demonstrated in the previous papers [4-7],

the CE/SE scheme features:

(1) space and time treated on the same footing, the integral equations of conservation laws are solved for with

second order accuracy,

(2) high resolution, low dispersion and low dissipation,

(3) novel, truly multi-dimensional, simple but effective non-reflecting boundary condition,

(4) Straight forward implementation of computation,

no numerical fix or parameter choice is needed;

(5) robust enough to cover a wide spectrum of compressible flow: from weak linear acoustic waves to

strong, discontinuous waves (shocks), appropriate for

linear and non-linear aeroacoustics.

Currently, the CE/SE scheme has been developed to

such a stage that a 3-D unstructured CE/SE NavierStokes solver is already available. However, in the

present paper, as a general introduction to the CE/SE

method, only the 2-D unstructured Euler CE/SE solver

is chosen as a prototype and is sketched in Section 2.

Then application of the CE/SE scheme to the auto-door

gap noise problem is depicted in Sections 3.

2 The 2-D Unstructured CF_JSE Scheme

The CE/SE scheme can be used with either structured

or unstructured grids. Here, the unstructured CE/SE is

used as a prototype for introductory description.

2.1 Conservation Form of the Unsteady Euler

Equations

Consider a dimensionless conservation form of the unsteady Euler equations of a perfect gas. Let p, u, v,

p, and "7 be the density, streamwise velocity component, transversal velocity component, static pressure, and

constant specific heat ratio, respectively. The 2-D Euler equations then can be written in the following vector

form:

Ut +Fx + G_ = 0, (1)

where z, 71, and t are the streamwise and transversal coordinates and time, respectively. The conservative flow

NASA/TM--2001-210703 1
variablevectorUandthefluxvectorsFandG,aregiven

by:

iv2 F2

v=/_ q , F= F_ ' G=

\ U4 F4

G2

G3 '

\G4

with

UI = p, _ = pu, &_ = pt,, _q = p/(_,-1)+p(u2+v2)/2;

F_ = U2, F.3 = UW31_q,

F,. = (_ - 1)u, + [(3- ._TT'_, ,_'2 -- (_ -- 1)U_]/2U,,

F.I -- _t'_[q/Ul - ('3- 1)U2 [_ + U_]/2U?,

G_ = b3, G2 = U2U3/_q,

c;,_= (_ - 1)_q + [(3- _)_r__ (_,- liVe]/2v1,

By considering (x,y, t) as coordinates of a threedimensional Eucfidean space E3 and using Gauss' divergence theorem, it follows that Eq. (1) is equivalent to the

following integral equation:

_s = 0, m = 1,2, 3, 4, (2) Hm dS

(v/

where S(V) denotes the surface around a volume V in

E_ andHm = (Fr_,Gm, Urn),m = 1,2,3,4.

2.2 Unstructured CE/SE

The CE/SE scheme is naturally adapted to unstructured

triangle grids. Each triangle center (O in AABC )

and its 3 neighboring triangle centers (D, E, F ) form 3

conservation elements or CEs (quadrilateral cylinders -

ADBO, BECO and CFAO), as shown in Fig. 1. These

triangle centers are the nodes where the unknowns U,

U,,, Uy are defined.

Assume that at the previous time level n (Fig. 1), U,

and its spatial derivatives U.r, Uy are given at all the

nodes (triangle centers), the CE/SE time marching is

based on a 'tripod' mode, i.e., U, Uz, U u at the new

time level n + 1 (e.g., the shaded circle & in Fig. 1) are

computed from their data at its surrounding neighboring

nodes at time level n (e.g., D in Fig. 1, note that O itseff

is not used).

During the time marching, the above flux conservation relation (2) in space-time is the only mechanism that

transfers information between node points. A conservation element CE, or computational cell, is the finite volume to which (2) is applied. Discontinuities are allowed

to occur in the interior of a conservation element and

nowhere else. A solution element SE associated with

a node (e.g., D, E, F in Fig. 1) is here a set of interface

planes in E3 that passes through this node (e.g. planes

DAA'D', DBB'D', DBOA associated with node D).

Within a given solution element SE(j, n), where j, n

are the node index,and time step respectively, the flow

variables are not only considered continuous but are also

approximated by linear Taylor expansions:

u*(_, u,t_j, n) = u_' + (ux)_(_ - xj)+

(u_)y(y - uj) ÷ (u,)Tft - r'), (3)

F*(x,y,t;j,n) = F__ + (Fx)y(:r -xj)+

(F._)j" (y - yj) + (F,)_(t - r'), (4)

G*(x,y,t;j,n) = Gy + (G_)}_(x - xj)÷

(Gu)y(y - yj) + (Gt)y(t - tn), (5)

where j is the node index of D, E or F, the partial

derivatives of F and G can be related to the corresponding one of U by using the chain rule and Ut can be

obtained from (1). These Taylor expansions are used

to accurately evaluating fluxes on SE_s e.g., on planes

DAAID ', DBB'D' or DBOA.

In principle, the number of equations derived from

these flux conservation laws (each CE provides 4 scalar

equations and the total number of scalar equations is

12) matches the number of unknowns (here 12 scalar

unknowns). All the unknowns are solved for based on

these relations. No extrapolations (interpolations) across

a stencil of cells are needed or allowed.

In practice, there is no need to solve a 12 × 12 equation system. In the space-time E3 space, (2) is applied to

the hexagon cylinder ADBECF - A'D'B'E'C_F '. U

at the hexagon center at the new time level is first evaluated and then, U at the center & of triangle A'B_C '

is obtained by Taylor expansion. Since the evaluation

of U_ and U u involves application of artificial damping

and some form of limiters (weighted average), the reader

is referred to the original papers [ 1-3] of Chang et al for

details.

2.3 Non-Reflecting Boundary Conditions

In the CF_JSE scheme, non-reflecting boundary conditions (NRBC) are constructed so as to allow fluxes from

the interior domain to smoothly exit to the exterior of

the domain. Some variants of the NRBC frequently employed in CE/SE schemes are:

(1) Type I - 'steady NRBC':

For a ghost grid node (j, n) lying outside the domain

at the top (or bottom) of the domain the NRBC requires

that

U n U n ( _)_=( _)i=°,

NASA/TM--2001-210703 2
,,,,,'_ .._,_._,. _.... _....... _" . rB

(qu_Lvilateral s)

C' E'

time level_+lA, F_

B'

I , II :\D' ,dj_ 1

! "r_l !

i I ii !

! i i, !

!_ I i!c I_E

A _-E[_ "" - _-_Q-"

time level _"_1----_ B

hexagon cyliner ADBECFA - A'D'B'E'C'F'A' end

its 3 CEI, OADB-O'A'D'B', OBEC-O'B'E'C',

OCFA - O'C'F'A'

Figure 1: CE/SE unstructured grid, showing 3 CEs and

the hexagon cylinder in E3.

while U n is kept fixed at the initially given steady bound- 3

ary value.

(2) Type II - 'outflow NRBC':

At the downstream boundary, where there are substantial gradients in y direction, the NRBC requires that

U n n : lTn-l/2 n (U ]n-1/2 ( x)j =0, Uj _j, (Uy)j =, _ ,

where j' is the index of an interior node closest to the

boundary ghost node j and U n and (U y)j'_ are now de- J

fined by simple extrapolation from the interior. This

NRBC is valid for either supersonic or subsonic flows.

It should be noted that although these NRBC's bear similarity to those used in finite difference schemes, the role

they play is very different. It can be shown that the above

NRBC's allow fluxes to smoothly exit to the exterior of

the domain.

3 Numerical results

Figure 2 illustrates the geometric configuration of

the door gap noise problem, which represents a typical aeroacoustic feedback system, producing sustainable

Air flow

/ door

/ C C___gap

,,",/////_ _ 3I/<//

A=15.9mm, B=28mm, C=8.76mm,

D=3.3mm

Figure 2: auto-door gap problem and the gap dimensions.

feedback cycle oscillations as a result of vortex shedding. The problem is given as a benchmark problem

at the 3rd CAA (Computational Aeroacoustics) Workshop (NASA Glenn ,Cleveland, OH, November, 1999)

and the corresponding experimental data will be released

later. The actual dimensions of the auto-door gap cavity

are also prescribed in Fig. 2. The length scale is chosen

to be twice the dimension C in Fig. 2, and the speed of

sound and density of the ambient flow are respectively

the scales for velocity and density. In the current computation, there are 48184 triangle elements for the unstructured mesh in the computational domain. These triangles are actually obtained by dividing a rectangular structured mesh cell into 4 pieces. The rectangular cell keeps

a uniform size of Ax = 0.00625 and Ay = 0.01136

around the area of the gap and the interior of the cavity,

but grows larger near the outer boundaries. At the gap

and cavity solid walls, no slip boundary conditions are

applied.

3.1 High speed case (v = 50.9res)

The mean flow follows the x-direction with Mach number M = 0.1497, corresponding to the car speed of

50.9m/s. Initially, ambient flow condition is imposed

on the entire flow field. At the inflow, the M = 0.1497

flow is imposed. At the top and bottom the Type I NRBC

is used while at the outflow, the Type II NRBC is specified. The domain shown in Fig. 3 is exactly the the computational domain, no buffer zone is used but still, the

CE/SE NRBC works well. Fig. 3 is taken at time steps

4000, 14000, and 24000 (with At = 0.00125), it shows

isobar snapshots of the computed flow field at different

stages. The generation of vortices and the generation of

NASA]TM--2001-210703 3
t=5 t=17.5 t=30

Figure 3: Isobar snapshots at different time steps.

(nonlinear) acoustic waves by vortex- gap edge impingement, both inside and outside of the cavity, are clearly

displayed.

The computation is totally carried out for 225,000

time steps. Starting from time step 60,000, when the unsteady flow is considered as fully developed, the pressure

history at the mid-point of the left wall of the cavity is

recorded and provided later for FFT (fast Fourier transform) analysis.

Fig. 4 displays the PSD (power spectrum density) in

log. scale for the time series. The x- axis denotes the

reduced frequency - the Strouhal numberSt. St = 1

is equivalent to a frequency of 19,406 Hz. It is observed that there are several different tone spikes extending to very high frequencies. The frequency at the lowest

tone is about 1,839 Hz. Despite the coarse resolution in

frequency domain, this agrees well with the experimental frequency data 1,824 Hz provided by the 3rd CAA

Workshop [8].

Fig. 5 is an enlargement of the numerical Schlierens

around the cavity at time step 225,000. Eddies are

clearly displayed.

3.2 Low speed case (v = 26.8res)

In this case, all the conditions are the same as above except the mean flow Mach number is lower, M : .0788,

corresponding to the car speed of v = 26.8m/s. For running at such a low Mach number, most of the numerical

schemes need some form of preconditioning. However,

this doesn't seem to bother the CE/SE scheme. Totally

390,000 time steps are run for this case and the PSD is

shown in Fig. 6. The strongest (also the lowest) tone frequency is estimated to be about 1,911 Hz. Since this

frequency is not far away from 1,824 Hz, while the car

speed is almost reduced by half, it is guessed that the tone

may come from the Helmholtz resonant frequency of the

cavity.

4 Concluding Remarks

A typical vortex induced, self-excited tonal gap noise

is simulated and compared to the experimental data.

Through the 'difficult' problem, the capability of the new

CE/SE scheme is demonstrated. Specifically,

1. the novel non-reflecting boundary conditions based

on flux balance is simple, genuinely multidimensional, and easy to implement, it is effective

even for near field boundaries;

. the (2nd order) CE/SE scheme is robust, efficient

and yields high resolution, low dispersion results

similar to those of higher-order schemes.

NASA/TM--2001-210703 4
conv.psd

-7.5

-11.0

-14.5

-18.0

0.0 2.0 4.0 6.0 5.0

13 Sep O0 10:37:01

Figure 4: PSD at a node inside the door cavity; x-axis: Strouha] number, high speed case.

conv.psd

-7.5

-10.0

-12.5

-15.0

0.00

I

i

i

i i

i

i

i

i

I

I b I I

0.50 1.00 1.50 2.00

5 Dec 00 13:13:55

Figure 6: PSD at a node inside the door cavity; x-axis: Strouhal number, low speed case.

NASA/TM--2001-210703 5
Figure5:Numerical Schlierensinthecavityarea.

References

[1] Chang, S. C., "The Method of Space-Time Conservation Element and Solution Element--A New

Approach for Solving the Navier-Stokes and Euler

Equations," Journal of Computational Physics. v.

119, 295-324 (1995).

[2] Chang, S.-C., Wang, X.-Y. and Chow, C.-Y., "The

Space-Time Conservation Element and Solution

Element Method: A New High Resolution and

Genuinely Multidimensional Paradigm for Solving

Conservation Laws ", J. Comp. Phys. v. 156, 89-136

(1999).

[3] Wang, X.Y. and Chang, S.C. "A 2D Non-splitting

Unstructured Triangular Mesh Euler Solver Based

on the Space-Time Conservation Element and Solution 'Element Method ", CFD Journal,vol. 8,

pp.309-325, 1999. (1998).

[4] Loh, Ching Y., Huitgren, Lennart S. and SinChung Chang, "Computing Waves in Compressible

Flow Using the Space-Time Conservation Element

Solution Element Method," AIAA Paper 98-0369

(1998), also to appear in AIAA Journal.

[5] Loh, C. Y., Hultgren, L. S., Chang, S.-C. and Jorgenson, P. C. E., "Vortex Dynamics Simulation in

Aeroacoustics by the Space-Time Conservation Element Solution Element Method," AIAA Paper 99-

0359 (1999).

[6] Loh, C. Y., Hultgren, L. S., Wang,Xiao Y., Chang,

S.-C. and Jorgenson, P. C. E., "Aeroacoustics Computation for Nearly fully Expanded Supersonic jets

Using the CE/SE Method", AIAA Paper 2000-2010

(2000).

[7] Loh, C. Y., Wang,Xiao Y., Chang, S.-C. and Jorgenson, P. C. E., " Computation of Feedback Aeroacoustic System by the CF_JSEMethod" Presented at

the 1st International CFD Conference, July, 2000,

Kyoto, Japan.

[8] "The Third Workshop on Benchmark Problems in

Computational Aeroacoustics", OAI/NASA Glenn,

Ohio, U.S.A. (Nov.,1999), in press.

NASA/TM--2001-210703 6

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GAP Noise Computation by the CE/SE Method

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Ching Y. Loh, Sin-Chung Chang, Xiao Y. Wang, and Philip C.E. Jorgenson

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NASA TM--2001-210703

FEDSM-2001-18136

Ching Y. Loh and Xiao Y. Wang, Taitech, Inc., 21000 Brookpark Road, Cleveland, Ohio; Sin-Chung Chang and Philip

C.E. Jorgenson, NASA Glenn Research Center. Prepared for the 2001 Engineering Division Summer Meeting sponsored

by the American Society of Mechanical Engineers, New Orleans, Louisiana, May 29-June 1, 2001. Responsible person,

Ching Y. Loh, organization code 5880, 216--433-3981.

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13. AB:_T_ACT (Maximum 200 words)

A typical gap noise problem is considered in this paper using the new space-time conservation element and solution

element (CE/SE) method. Implementation of the computation is straightforward. No turbulence model, LES (large eddy

simulation) or a preset boundary layer profile is used, yet the computed frequency agrees well with the

experimental one.

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Gap noise; CE/SE method

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