Dispersion-Relation-Preserving Interpolation Stencils. for Computational Aeroacoustics

Transcription

1 Adv. Theor. Appl. Mech., Vol. 3, 00, no., Dispersion-Relation-Preserving Interpolation tencils for Computational Aeroacoustics Zhanxin Liu Yichang Testing Technique Research Institute, o.58 hengli 3 rd Road Yichang, Hubei , China hustliuzx@gmail.com Abstract: From the example of investigating the mechanism of sound wave propagation at the inlet of a jet engine, we find that during the computation there inevitably exist some points whose values cannot be computed directly or as accurate as at other points. Interpolation method is a good way to obtain the values of these points from the others that can be accurately computed. To solve aeroacoustics problems, the interpolation stencils must have the properties of dispersion-relation-preserving (DRP). And the --point stencils are optimized by Lagrange multiplier method under the concept of DRP. The optimized DRP interpolation stencils are more accurate than Lagrangian interpolation ones in the small wave number range and they are successfully used in the example. The method described in this paper can also be used to optimize other interpolation or extrapolation stencils Keywords: interpolation; dispersion relation preserving; computational aeroacoustics; inlet; jet engine Introduction owadays, computational aeroacoustics (CAA) pays a more and more important role in the design of modern aircrafts. For example, Monola[] and Iansteel [] used CAA methods to investigate noise propagation at the inlet of a jet engine; Tam etc. [3,4]predict jet noise by CAA methods. The objective of CAA is not just to develop computational methods but also to use these methods to solve

2 5 Zhanxin Liu real practical aeroacoustic problems, so now more and more researchers are interested in CAA methods and its applications. There are too many differences between computational fluid dynamics (CFD) and CAA[5], one of those differences is that we should not only consider the aeroacoustic problems in time domain but also in frequency domain. Tam [6] first pointed out that discretization finite difference equations (FDE) must preserve the same dispersion relation with that of the original partial differential equations (PDE) in the small wave number range. He derived this important dispersion relation preserving (DRP) concept by analyzing the deviations of group velocity between PDE the FDE which are the reason of numerical dispersion. Afterward, Hu [7] optimized Runge-Kutta time integration method under the concept of DRP; Bogey [8] summarized all the explicit schemes and Kim [9] and Liu [0,] optimized compact difference schemes and filtering schemes. The methods for CAA not only include these temporal and special discretization schemes but also contain proper boundary conditions and other special stencils, for example, interpolation[]. In many occasions, some points can not be computed directly. For instance, when we solve governing equation in cylindrical coordinate, the points whose radial coordinate, r, is zero can not be computed since /r goes to infinite; when mesh the physical plane to computational plane by conformal mapping methods, there may be exist singular points, and the values of these singular points can not be computed directly as accurate as at other points. A good choice to obtain the values of these points is to interpolate them from the other points whose values can be easily got from direct computation. ince the discretization schemes have the property of DRP, the other special stencils must too have this property. Or else, the special stencils will deteriorate the whole computation accuracy. In this paper, we will adopt a new method to optimize the interpolation stencils under the concept of DRP described in []. We will first introduce the method and then compare the optimized results with Lagrangian interpolation methods, and present an example at last. point Interpolation stencil Profiles of the casing and spinner of a jet engine are shown in Figure. They are symmetric with respect to x- axis. This model is extensively used to investigate the mechanism of noise propagation at the inlet of a jet engine, e.g., [,3]. When we do the computation by FD methods, interpolation is needed at two places.

3 Dispersion-relation-preserving interpolation stencils 53 Figure. Profiles of the casing and spinner of a jet engine. ince this model is symmetric, we only need to do the computation in the upper plane. The points on the axis of symmetry are interpolated instead of being computed directly since the r = 0 and /r goes to infinite. The interpolation stencil for these points is shown in Figure. This is a -3-point symmetric stencil where 3 source points whose values are computed directly are located on the left and 3 others on the right. Figure Interpolation stencil at the axis of symmetry. (C-D is the axis of symmetry; point A is the one whose value is interpolated by the other 6 points) All the computation are done in the computational plane, so first we need to transform these profiles into computational plane to make sure their images are nearly lines by conformal mapping method. During the mapping 3 singular points which are located at the tip of the profiles are inevitably introduced. To ensure the mesh around the singular point is fine, overset girds method is applied. The mesh around the singular point of casing in physical plane is shown in Figure 3. These grids are first meshed in computational plane in elliptic coordinate and then transformed into physical plane inversely by ewton s iteration method. We can see that there are some wiggles near the singular point and this should not be

4 54 Zhanxin Liu allowed since the profile of casing is very smooth. These wiggles are resulted from the numerical errors of the ewton s iteration method. We can smooth out the physical coordinates of these points by interpolation from the other points which can be accurately computed in computational plane. The interpolation stencil is shown in Figure 4. The interpolation stencils shown in Figures and 4 could be generalized into a stencil which is illustrated in Figure 5 and expressed as where Δx j [ ] l [ ] () f ( x) = f x+ ( j+ m) Δ x + f x+ ( l + n) Δx j= l= is the uniform spatial interval. For the stencil in Figure, = 3; m= n=. For the stencil in Figure 4, = 6; m, n=, 4 or mn=,,3. Figure 3 Mesh around the singular point of casing in physical plane (The point in filled circle is singular ). Figure 4 Interpolation stencil for the points around the singular points (points B-B4 are interpolated by the numbered points) Figure 5 Generalize point interpolation stencil.

5 Dispersion-relation-preserving interpolation stencils 55 3 Optimization method Let ( x + ) i f( x) = Ae α ϕ and substitute it into Equation (). The local error for a givenα can be expressed as iϕ i( j+ m) αδ x+ iϕ i( l + n) αδ x+ iϕ local = j l j= l= () E e e e urely, we need to keep the local error small in the small wave number range. o let s defined the global error as By substituting Eq. () into (3), we have E = Elocald( αδx) (3) 0 i( j+ m) αδx i( l + n) αδx j l ( α ) 0 j= l= E = e e d Δx (4) The interpolation error should be zero if the function is a constant orαδ x = 0, that is to say, + l = (5) j j= l= ow, the coefficients of interpolation formula are to be chosen such that E is a minimum subject to constraint (5). If is given, the error E is a function of a single independent variable. The constrained optimized problem can be solved by the method of Lagrange multiplier. The Lagrangian function may be defined as, i( j+ m) αδx i( l + n) αδx j l α μ j l j= l= j= l= L= e e d( Δ x) + + (6) 0 The coefficients and the Lagrange parameter μ are found by solving L L L = 0, = 0, = 0 μ j l (7) By doing some algebra, Eq. (7) falls into

6 56 Zhanxin Liu L= jcos( j+ m) β+ lcos( l + n) β dβ+ jsin( j+ m) β+ lsin( l + n) βdβ+ μ j+ l 0 j= l= 0 j= l= j= l= (8) where β = αδ x, then L k sin( j k) sin( l+ n k+ m ) sin( k+ m) j + l + μ for k j= j k l= l+ n k+ m k+ m = sin( j m k+ n+ ) sin( l l) sin( k + n) j + l + μ for k j= j m k+ n+ l= l k k + n (9) ubstituting Eq.(9) into (8), we have sin( j k) sin( l+ n k+ m ) sin( k+ m) μ j + l + =0 for k j= j k l= l+ n k+ m k+ m sin( j m k+ n+ ) sin( l l) sin( k + n) μ j + l + = 0 for k j= j m k+ n+ l= l k k + n (0) On the other hand, L / μ = 0only implies Eq.(5). Combining Eq (0) and (5), we can find the coefficients by solving the system Ax = b () where x and b are vectors with a length +, A is a matrix of ( ) ( ) + +. And k for - k, k 0 x k+ + = μ for k = 0 When k sin( q k) for q q k A k+ +, q+ + = / for q= 0 sin( q+ n k+ m ) for q q+ n k+ m When k = 0

7 Dispersion-relation-preserving interpolation stencils 57 A k+ +, q+ + for q, q 0 = 0 for q 0 When k A b k+ +, q+ + k+ + sin( q m k+ n+ ) for q q m k+ n+ = / for q= 0 sin( q k) for q q k sin( k+ m) for k k+ m = for k = 0 sin( k + n) for k k + n, When = 3, m= n=, A,x,b are expressed as sin sin 4 sin 5 sin 6 sin sin 3 sin 4 sin 5 sin sin sin sin sin 3 sin 4 sin 3 4 A = 0 sin 4 sin 3 sin sin sin 4 3 sin 5 sin 4 sin 3 sin sin sin 6 sin 5 sin 4 sin sin x = μ, 3

8 58 Zhanxin Liu sin 3 3 sin sin b = sin sin sin 3 3 By solving this liner system, we can easily find the optimized coefficients for the interpolation stencils for given. 4 Optimized results During the optimization, we set =. which is the same with that of DPR 7-point FD schemes. The optimized results are tabulated in Table. The two cases of = 6, m=, n= 4 and m= 3, n=, are related by ( m=, n= 4) = ( m= 4, n= ) j j ( m= 3, n= ) = ( m=, n= 3) j j () The local errors of the three cases in Table are shown in Figures 6-8, comparing with those of Lagrangian interpolation method. From the figures we can see that the local errors of the optimized ones are much smaller than those of Lagrangian ones in the small wave number region [ 0,.]. After interpolation by the optimized stencils, the hard wall around the singular point of casing become much smoother and it is shown in Figure 9.

9 Dispersion-relation-preserving interpolation stencils 59 Table. The optimized results of --point interpolation stencils = 6, m= 4, n= = 6, m= 3, n= = 3, m=, n= Figure 6. local error of interpolation stencil ( =6,m=4,n=; solid- optimized; dash- Lagrangian)

10 530 Zhanxin Liu Figure 7. local error of interpolation stencil (=6,m=3,n=;solidoptimized; dash- Lagrangian) Figure 8. local error of interpolation stencil ( =3,m=,n=; solid- optimized; dash- Lagrangian) Figure 9. Mesh around the singular point of casing in physical plane after smoothness by interpolation (The point in filled circle is singular).

11 Dispersion-relation-preserving interpolation stencils 53 5 Example To demonstrate the successful optimization, we will simulate the noise propagation at the inlet of a jet engine which is shown in Figure. The sound wave for this axis symmetric problem is governed by linearized Euler equation which can be expressed as ρ ρ ρ ρ ρ v v w u v v u + v + v + u + u + ρ ρ + + = 0 t r r x x r r r φ x r r x u u u u u p ρ p + v + v + u + u + = 0 t r r x x ρ x ρ x v v v v v p ρ p + v + v + u + u + = 0 t r r x x ρ r ρ r w w w vw p + v + u + + = 0 t r x r ρ r φ p p p p p v v w u v v u + v + v + u + u + γ p γ p + + = 0 t r r x x r r r φ x r r x (3) where variables ρ, uvwp,,, represent density, velocity at x-direction, velocity at r-direction, velocity at φ (rotation )direction, pressure, respectively; the quantities with prime represent acoustic variables and the ones with bar represent mean flow variables. In order to simplify this 3-dimensional problem to a -dimensional one, we assume the sound waves at the inlet have analytical spinning modes which take the form of ρ ˆ( ρ rxt,,) u uˆ( r, x,) t imφ v = Re vˆ (, r x,) t e w wˆ (, r x,) t p pˆ( r, x,) t (4) where i =. By substituting Eq.(4) to (3), it follows

12 53 Zhanxin Liu ˆ ρ ˆ ρ ρ ˆ ρ ρ vˆ vˆ im uˆ ˆ ˆ ˆ ˆ v v u + v + v + u + u + ρ + + w+ + ρ + + = 0 t r r x x r r r x r r x uˆ uˆ u uˆ u pˆ ˆ ρ p + v + vˆ + u + uˆ + = 0 t r r x x ρ x ρ x vˆ vˆ v vˆ v pˆ ˆ ρ p + v + vˆ + u + uˆ + = 0 t r r x x ρ r ρ r wˆ wˆ wˆ vwˆ im + v + u + + pˆ = 0 t r x r ρ r pˆ pˆ p pˆ p vˆ vˆ im uˆ v v u + v + vˆ + u + uˆ + γ p + + wˆ + + γ pˆ + + = 0 t r r x x r r r x r r x This governing equation is computed in the domain shown in Figure 0. At the far field boundaries, A-B-C-D in Figure 0, outflow boundary conditions[6] are imposed. At the hard wall of casing and spinner, slip boundary conditions are imposed by ghost point methods. Perfectly matched layer (PML) boundary conditions [4,5] are used in the region E-F-G-H to prescribe the sound wave at the fan face. At the line A-K, axial boundary conditions are imposed by the optimized -3-point interpolation method.. Two cases are computed, both of the inputted sound waves has a frequency of 63 Hz and a sound power lever of 30 decibel. The first one has a azimuthal mode m = 3 and a radial mode n =. The second case has the same azimuthal mode but the radial mode is 4. During the computation, all the variables are dimensionless by the following unit scale, Length = D( the inner diameter of casing) Velocity = a 0 (sound speed in standard atmosphere) (5) Density = ρ 0 (standard atmospheric density) Time = D/ a 0 Pressure = ρ a 0 0 The pressure contours at time t = 30 of the two cases are shown in Figures and, respectively. From the figures we can see the propagation process of sound wave at the inlet. ince the examples are only used to demonstrate the optimized interpolation stencils, the detailed comparison and analysis of the sound wave propagation are not presented in this paper.

13 Dispersion-relation-preserving interpolation stencils 533 Figure 0 The domain for mean flow computation. Figure. Pressure contour at t=30 of m=3,n= Figure. Pressure contour at t=30 of m=3,n=4

14 534 Zhanxin Liu 6 Conclusion ) During the computation, there inevitably exist some points whose values can not be computed directly or can not be calculated as accurate as at other points. Interpolation is a good way to obtain the values or to smooth out numerical errors. ) ince the discretization methods for both time and space have the property of DRP, the interpolation stencils must also have this property to keep the accuracy of the whole computation. The coefficients of the interpolation stencils can be optimized by Lagrange multiplier methods. The optimized DRP interpolation stencils are much more accurate than Lagrangian interpolation ones in the optimization wave number range. 3) The examples show not only the success of imposing axis symmetric boundary conditions and smoothing the errors around the singular point by DRP interpolation stencils but also the power of DRP FD method to solve aeroacoustic problems. References [] Manoha E, Mincu DC. umerical simulation of the fan noise radiated through a semi-buried air inlet. AIAA paper 009: [] Iansteel Achunche JA, Rie ugimoto. Prediction of Foward Fan oise Propagation and Radiation from Intakes AIAA paper 009: [3] Tam CKW, Pastouchenko, Viswanathan K. Broadband shock-cell noise from dual stream jets. Journal of ound and Vibration 009; 34: [4] Tide P, Babu V. umerical predictions of noise due to subsonic jets from nozzles with and without chevrons. Applied Acoustics 009; 70: [5] Tam CKW. Computational aeroacoustics: An overview of computational challenges and applications. International Journal of Computational Fluid Dynamics 004; 8: [6] Tam CKW, Webb JC. Dispersion-relation-preserving finite difference schemes for computational acoustics. Journal of Computational Physics 993; 07: 6-8. [7] Hu FQ, Hussaini MY, Manthey JL. Low-dissipation and low-dispersion Runge-Kutta schemes for computational acoustics. Journal of Computational Physics 996; 4: [8] Bogey C, Bailly C. A family of low dispersive and low dissipative explicit schemes for flow and noise computations. Journal of Computational Physics 004; 94: 94-4.

15 Dispersion-relation-preserving interpolation stencils 535 [9] Kim JW. Optimised boundary compact finite difference schemes for computational aeroacoustics. Journal of Computational Physics 007; 5: [0] Liu Z, Qibai H, Li H, Jixuan Y. Optimized compact filtering schemes for computational aeroacoustics. International Journal for umerical Methods in Fluids 009; 60. [] Liu Z, Huang Q, Zhao Z, Yuan J. Optimized compact finite difference schemes with high accuracy and maximum resolution. International journal of aeroacoustics 008; 7: [] Tam CKW, Kurbatskii KA. A wavenumber based extrapolation and interpolation method for use in conjunction with high-order finite difference schemes. Journal of Computational Physics 000; 57: [3] Özyörük Y, Alpman E, Ahuja V, Long L. Frequency-domain prediction of turbofan noise radiation. Journal of ound and Vibration 004; 70: [4] Hu FQ. Development of PML absorbing boundary conditions for computational aeroacoustics: A progress review. Computers and Fluids 008; 37: [5] Tam CKW, Ju H, Chien EW. cattering of acoustic duct modes by axial liner splices. Journal of ound and Vibration 008; 30: Received: eptember, 00