Gridless Solution Method for Two-Dimensional ... - CyberLeninka

V ol. 18

No. 1

CH INES E JO U RN A L O F AER ON A U TICS

February 2005

Gridless Solution Method for Two Dimensional Unsteady Flow WANG Gang , SUN Ying dan, YE Zheng yin ( College of A er onautics , Nor thw ester n Poly technic University , X i an Abstract:

710072 , Chi na)

T he main purpose of this paper is to develop a g ridless method for unsteady flow simulat ion.

A quadr antal point infilling strategy is developed to g enerate point and combine clouds of po ints automati cally. A point moving algor ithm is introduced to ensure the clouds of points follow ing the movements of bodyboundaries. A dual time method for solving the tw o dimensional Euler equations in Arbitrary La gr angian Euler ian ( A LE) formulation is presented. Dual time method allows the real time step to be chosen on t he basis of accuracy rather t han stability. It also permits the acceleration techniques, w hich are co mmonly used to speed up steady flo w calculations, to be used w hen marching the equations in pseu do time. T he spatial derivatives, w hich ar e used to estimat ing the inviscid flux, are dir ectly appr oximat ed by using local least squares cur ve method. A n ex plicit multistage Rung e K utta algorithm is used to ad vance the flow equations in pseudo time. In order to accelerate the solution to convergence, local time stepping technique and residual averaging are employed. T he r esults of N ACA 0012 airfoil in transonic steady flow are pr esented to verify the accur acy of the present spatial discretizat ion met hod. Finally, tw o A GARD standard test cases in which NACA0012 airfoil and NA CA64A010 airfoil oscillate in transonic flow are simulated. T he computational results ar e compared wit h the ex perimental data to demonstrate t he validity and practicality of the presented method. Key words:

computational fluid dynamics; gr idless method; dual time method; unsteady flow ; Euler

equat ion 一种计算 非定常二维流动的无网格算法. 王

刚, 孙 迎丹, 叶正寅. 中国航 空学报 ( 英 文版) , 2005,

18( 1) : 8- 14. 摘 要: 主要目的是发展一套求解 非定常 流动的 无网格 算法。计算 区域的 离散方 面, 提 出了一 种 按区域进 行填充布点的点云自动生成方法; 发展了一种点云的 运动技术 来实现离散 点云对物 面边 界的随体运动; 在点云离散的基础上, 采用最小二乘法求解矛盾方 程的方法 来求取空间 导数, 进而 获得数值通量; 采用双时间方法进行时间离散推进, 其中物理时间 迭代采用 二阶隐式格 式, 伪 时间 迭代采用 四步龙格- 库塔显式格式, 为了加 速收敛, 在伪 时间迭 代中采 用了当 地时间步 长和隐 式 残值光顺 等加速收敛措施。最后, 利用 本文 算法模 拟了 NA CA0012 翼型 和 N ACA 64A 010 翼型 的 跨音速非定常流动, 并将计算结果与 实验测 量结果 进行了对 比分析, 验 证了上 述方法的 正确性 和 实用性。 关键词: 计算流体力学; 无网格方法; 双时间方法; 非定常流; Euler 方程 文章编号: 1000 9361( 2005) 01 0008 07

中图分类号: V211. 3

According to t he discret izat ion manners of comput at ional region,

文献标识码: A

local least squares curve f it s over each cloud of

numerical algorithms in

points. Gridless met hods have much geomet rical

comput at ional f luid dynamics ( CF D) can be divid ed into t w o rough sorts: t he classical g rid methods

flex ibility for computing wit h complex configura t ions because they throw of f t he grid cell t houg ht

[ 1 5]

and the burgeoning gridless methods in recent years. In gridless methods, clouds of points w hich

and break aw ay from t he restrict ions of t he grid quality and topolog ical st ruct ure.

are composed of the point itself and its neighbor points, are adopt ed to subst itut e f or g rid cells. T he

During using g rid algorit hm to calculate t he unst eady f low , t he body fit ted dynam ic grid tech

spat ial derivatives are direct ly det ermined by using

nique

[ 6 8]

must be used to simulat e unsteady mo

R eceived dat e: 2004 04 21; R evision received dat e: 2004 07 31 Foundation it em: The D oct orat e Creat ion Foundat ion of N orthw est ern Polytechnic U niversit y

February 2005

G ridless Solution M et hod for T wo Dimensional U nst eady Flow

9

t ions of body boundaries. But w ith the restrict ions

process, spat ial derivatives of point i are evaluated

of the grid qualit y and the grid topological struc

by a least squares f it algorithm in t he point cloud of

t ure, it is very dif ficult to use moving grids method

point i . T he f ollowing is t he basic idea of t he tech

in t he condit ions of body boundaries moving in large ex tent . As ment ioned above, gridless met h

nique for generat ing clouds of discrete points aut o mat ically .

ods have no rest rictions of grid skew ness and st ret ching dist ortions, thus, they have distinct ad vant ages for unst eady flow calculat ion. How ever, previous researches on gridless solut ion methods w ere m ainly concerned w it h steady f low . T he main purpose of this paper is t o develop a g ridless method f or unsteady flow simulat ion. A Fig. 1

quadrant point inf illing st rat egy is developed t o g enerat e point and combine clouds of points aut o mat ically . A point moving algorit hm is int roduced to ensure t he clouds of po ints follow ing the move ment s of body boundaries. T he spat ial derivat ives, w hich are used to est im at ing the inviscid flux , are direct ly approx im ated by using local least squares curve m et hod. A dual t ime met hod[ 7] for solving

Sketch map for the fr amewor k of the cloud of point i

F irst , all kinds of t he boundaries ( body boundaries and far field, etc) in t he f low computa t ional f ield are compart ment alized int o the form of discrete po ints . For an arbitrary discrete point i , according to t he given background informat ion, t he searching radius r of discret e point i in t he f low comput at ional field can be ascertained. T hen, t he

the t wo dimensional Euler equat ions in Arbitrary Lagrang ian Eulerian ( ALE) f ormulat ion is present

searching region for t he cloud of t he point i can be

ed. In t his approach, an implicit second ordert ime

def ined as a circle w it hin t he radius of r from t he point i . Especially , t he searching region is a fan

discret isation is used in the physical t ime marching procedure, the pseudo t ime integ rat ion is carried out by an explicit four step Runge Kut ta scheme and is accelerat ed by local t ime stepping and im plic it residual smoot hing. T he results of NACA0012 airfoil in transonic st eady flow are presented to ver ify t he accuracy of t he present spat ial discret izat ion met hod. F inally , tw o AGARD st andard test cas [ 9, 10]

es in w hich NACA0012 airfoil and NACA64A010 airfoil oscillat e in transonic flow are

shaped region if point i is on t he boundary. T he searching region of point i can be com part mental ized int o several sub regions on t he principle of keeping the f an shaped angles equivalent ( show n in Fig. 2) . Generally , if the point is in the flow com putat ional f ield, the searching reg ion is recom mended to be divided int o six sub regions, and if the po int is on t he boundary, f our sub regions are enoug h.

simulated. T he comput at ional result s are com pared w it h t he ex perimental dat a t o demonstrat e the va lidity and pract icality of the presented met hod.

1

Generation and M oving T echnique for Clouds of Points

In gridless met hod, each discrete po int in t he

Fig . 2

Sketch map for generating the cloud of point i by using quadrant po int infilling strategy

comput at ional f ield has it s ow n cloud of point s. As shown in Fig. 1, t he point cloud of point i is com

T hen, search t he point in each sub reg ion, if

posed of t he point i it self and it s neighbor point s

there have been already one or more discrete point s

( the points in the circle) . During t he flow solving

in a sub reg ion, t he point having t he short est dis

10

WA NG Gang, SU N Ying dan, YE Zheng yin

CJA

tance to point i should be added t o the cloud of

!v ( v - v P ) + p

point i , just like the solid point s show n in F ig . 2.

T he equat ions are non dimensionalized wit h a

If there is no point in a sub reg ion, a new point should be locat ed at t he site t hat is on f an shaped

reference densit y !! and a speed of sound a ! . Here !, u , v , p and e0 represent the g as density,

ang le bisect or of t his sub region and is 0 5 r aw ay

velocit y component s in x and y direct ions, pres

from t he point i . And then the new points ( hollow points show n in Fig 2 ) should be added t o t he

sure and tot al energy per unit volume. On t he per

cloud of point i . Repeat t he above process unt il t he

ed from the equation of state p = ( ∀- 1 )

cloud of every point in t he comput at ional f ield has been f ormed.

e 0( v - v P ) + p v } T

f ect gas assumpt ion, the pressure p can be calculat

e0 -

! 2 ( u + v 2 ) , w here ∀ is the rat io of spe 2

In unst eady f low com put at ion, in order t o

cif ic heat s of the f luid and typically t aken as 1 4 for

simulate the relative movement of body boundaries,

air, u P and v P represent velocity com ponents in x

it is request ed t hat clouds of discrete points have the ability of moving wit h t he body boundaries.

and y direct ions of t he discrete point s.

Hence, a point moving algorit hm is developed in the f ollowing descript ion. T ake point i in compu t at ional field as an example, it s displacement one time step is det ermined by d ij ri = rj sin D j

ri in

2. 2

Numerical discretization When gridless met hod is applied for t he spat ial

discret izat ion of t he governing equat ions, w here the shoe pinches is to determ ine the spatial deriva t ive of t he conserved variable Q. T he numerical

( 1)

w here the subscript j is t he index of boundary point which is t he nearest point aw ay from point i ,

F E + can be obt ained by t he derivat ive y x calculating principle of compound f unction. flux

Suppose t he value of !has been known on ev

d ij is t he distance f rom point i t o point j , Dj is t he

ery point , considering the cloud of a point show n in

shortest distance f rom point j to the ot her closed boundary curve and r j is the displacement of point

Fig. 1, t he spat ial derivat ives

j . It is easy t o determine rj during the course of comput at ion because point j is on t he boundary. In

number of discrete points in t he cloud of po int i ,

the sam e way, t he displacement of an arbitrary point in the f low field also can be determ ined as Eq. ( 1) .

2

Unsteady Flow Numerical Sim ulation M ethod

2. 1

! ! and can be x y comput ed by t he following method. If n is t he then a syst em of n - 1 linear equations w it h t he ! ! and can be obt ained by a x y first order accuracy T aylor series ex pansion about unknow n variables point i !j = !i +

! ( x - xi ) + x i j

! ( y - yi ) y i j

( j = 1, 2 ∀, n - 1)

Governing equations

( 3)

T he Euler equat ion in Arbitrary L ag rang ian Eulerian ( AL E) diff erent ial form for t he Cart esian

where t he subscript j is the index number of t he

coordinat e syst em can be described as f ollows:

point i . According to the const ruct ion manner of

Q t +

E x +

F y = 0

w here Q = ( ! !u

!v

E = { !( u - u P) !v ( u - u P) F = { !( v - v P )

T

e 0) ,

( 2)

discrete point s except for point i in the cloud of the cloud of points, the t otal number of discrete points in t he cloud should be more than three, hence the above equat ions are inconsist ent in fact.

!u( u - u P ) + p

In order to obt ain t he approximate solut ion of t he inconsistent Eq. ( 3) , a linear least squares method

e 0 ( u - u P) + p u} T ,

is used. It can be proved that t he algorithm of cal

!u( v - v P)

culat ing spat ial derivat ive in this paper is equiva

February 2005

11

G ridless Solution M et hod for T wo Dimensional U nst eady Flow 1 d Qn+ 1 / d#+ R *i ( Q n+ i ) = 0

lence to the met hod int roduced in Ref . [ 1, 3] .

( 7)

According to t he above met hod, the numerical

An ex plicit four stag e Runge Kut ta algorit hm

E F flux Ri = + in Eq. ( 2) can be derived x i y i from t he derivative calculat ing principle of com

is used to advance the flow equat ions in pseudo time, in which t he solution is advanced f rom pseu

pound f unction by using the spatial derivative of t he conserved variables !, !u, !v and e 0 on t he cloud of points. So t he semi discret ization f orm of Eq. ( 2) on point i can be ex pressed as

do t ime level n # t o level( n + 1) #, and can be writt en as Q(0) = Qni Q( m ) = Q (0) - ∃m #i R*i ( Q( m- 1) ) for m = 1 to 4 ( 8) n+ 1

d Qi + Ri = 0 dt

( 4)

Since the method of t he present w ork is essen t ially non dissipat ive, addit ional art if icial dissipa t ion t erms are required to prevent the t endency for

Qi

( 4)

= Q

where t he coef ficients ∃1 = 1/ 4, ∃2 = 1/ 3, ∃3 = 1/ 2, ∃4 = 1. T he major disadvant age of explicit schemes is that t he t ime st ep #i is restricted by the Courant Friedrichs Lew y ( CF L) st ability condit ion. In or

spurious even and odd point oscillat ions and t o en sure numerical stabilit y in t he course of spat ial dis

der t o accelerat e t he solut ion to convergence, local

cret ization. Follow ing the m et hod of Ref . [ 3] , ar

time stepping t echnique and implicit residual aver

t ificial dissipat ion terms are added t o Eq. ( 4) for

aging t echnique are employed in the pseudo t ime

point i , then it becomes

integrat ion.

d Qi + Ri= 0 dt

( 5)

w here R i = Ri - D i , Di are t he art ificial dissipa

given by the follow ing equat ion #i = min

t ion t erms. F or inviscid f low , the flow t ang ent condit ions at t he solid boundary points are imposed by set ting the slip velocit ies on t he boundary faces and elimi nat ing t he normal velocit y com ponent. In the far field, one dim ensional charact erist ic analysis based on Riemann invariants is used t o det ermine the val ues of the f low variables on t he out boundary of t he comput at ional dom ain. 2. 3

In a full implicit t ime concept , t ime deriva t ives in Eq. ( 5 ) are discret ized by second order backw ard difference method, t hen Eq. ( 5) be comes as 1 1 n+ 1 3 Qn+ - 4Q ni + Qn)= 0 i i / 2 t + R i ( Qi

( 6) the Eq. ( 6) in real t ime layer, t he derivative terms of conserved variables w it h respect to pseudo time, # is added to t he lef t side of Eq. ( 6) . By using t he 2 t+ R i (

1 Q n+ ) i

2 t 3

( 9)

where n- 1

C CF L #i = n - 1 # 1/ [ | u i / ( x i - x k ) | + | v i / ( y i k= 1 y k) | + ai

1/ ( x i - x k ) 2 + 1/ ( y i - y k ) 2 ] , C CFL

denot es the coeff icient of CFL , a i is t he local sound speed at point i , t he subscript k is the index num ber of discret e point s w hich belong t o the cloud of Residual averaging is anot her t echnique to ac celerat e the solut ion t o convergence. T his tech nique ext ends the reg ion of stabilization, and then increase the maximum value of t ime st epping per m it ted in t he pseudo time int egration. In gridless met hod, let R i represent the residual of point i , a new residual can be g iven by

A dual t ime met hod is introduced to advance

*

#i ,

point i .

Time integration

n+ 1

denot at ion of R i ( Q i

#i of discrete point i is

T he local t ime step

n+ 1

) = (3 Q i

n

n- 1

- 4Qi + Q i

, Eq. ( 6) can be w ritt en as

)/

N

R i = Ri+ %

# ( R k-

R i)

( 10)

k= 1

where %is the coefficient of residual averaging, and it s recommended value is a number bet w een 0 2 t o 0 5. T his t echnique of implicit residual averag ing allow s t he CF L number to be increased about tw o

12

WA NG Gang, SU N Ying dan, YE Zheng yin

CJA

times larger than t he unsmoot hed value.

3

Results

T he spatial discretizat ion accuracy of t he pre sent gridless method is evaluat ed by performing t he st eady flow calculat ion about transonic flow around the NACA 0012 airfoil. In order to furt her under st and t he charact eristics of t he current gridless met hod in t he case of unsteady f low , t he unsteady flow

simulat ions for

NACA0012

airf oil

Fig . 4

and

NACA64A010 airfoil oscillations in pit ch about quart er chord in t ransonic f low are conducted. For each test case, t he criterion of convergence for t he

Pressur e coefficient distr ibut ions for tr ansonic flow around N ACA0012 air foil

3. 2

Unsteady transonic flow around oscillating NACA0012 airfoil One of t he st andard test cases for t he

pseudo t ime int eg rat ion process is the m ax imum

NACA0012 airfo il is considered. T he com puted re

v alue of residual decrease to the order of 10- 6 .

sult s are compared wit h the classical AGARD ex

3. 1

Transonic steady flow around NACA0012 airfoil T o give an evaluat ion for numerical accuracy

[ 9]

periment dat a . In t his test case, t he NACA0012 airfoil oscillat es in pitch about it s quarter chord and the instant aneous angle of at tack ∃( t ) is given by

of t he present gridless method, a t ransonic steady

∃( t ) = ∃0 + ∃m sin ( &t )

flow at Ma = 0 80 and ∃ = 1 25∃ around NACA0012 airfoil is comput ed. T he close up view

where ∃0 is the mean incidence, ∃m is the amplitude of pitching angle, t he angular frequency & is related to

of t he f ield point s is displayed in F ig . 3. T here are

the reduced frequency k by the relationship k = &c/

( 11)

( 2 V ! ) , where c is the airfoil chord lengt h and V ! is the free stream speed. T he test case is simulated wit h the following conditions: Ma= 0 755, ∃0 = 0 016∃, ∃m = 2 51∃, k= 0 0814. T he dist ribut ion of cloud of points in t he f or mer t est case is taken as the init ial point dist ribu t ion of current case. T he com put at ion is carried out using 72 real time steps in one oscillation cycle. T he average run time per oscillat ing period is nearly Fig. 3 P ar tial view of po ints distr ibut ion around N ACA 0012 airfo il

200 point s distribut ed on the boundary of airfoil

0 3 h, and the number of pseudo time st eps for each real time st ep is approximate to 300. Com

and 6145 points distribut ed on the w hole computa

pared w ith t he ex perimental data, t he calculated instant aneous pressure coeff icient dist ribut ions at

t ional field. As show n in Fig. 4, the calculat ed sur

four point s in t im e during the f ort h circle of mot ion

f ace pressure coeff icient distribut ions compare w ell

are show n in F igs 5( a) 5( d) . In these f ig ures, ∋

w it h the experimental dat a. In t his t est case, both

( t ) denotes the phasic angle of oscillat ion at that

the st rong shock on the upper surf ace and t he weak

moment . Instant aneous lift coef ficients C l and in

shock on the lower surface are well capt ured.

st antaneous moment coef ficients C m vs the inst an

T herefore, t he gridless m et hod presented in t his paper has nicer capability to resolve f low discont i

t aneous angles of att ack are plot ted in Fig 6 and Fig 7. In t hese tw o figures, t he calculated result s

nuity.

of t he present approach agree w ell w it h the experi

February 2005

G ridless Solution M et hod for T wo Dimensional U nst eady Flow

ment al dat a and t he related numerical result s[ 6] .

13

present gridless met hod for unsteady f low simula t ion, the transonic f low around NACA64A010 air f oil oscillat ion in pit ch about its quart er chord is simulated. T he instant aneous angle of at tack can dist ribut ion of point s around t he NACA64A010 airfoil is also given by Eq. ( 11) and w it h the fol low ing conditions: Ma= 0 796, ∃0 = 0 0∃, ∃m = 1 01∃, k= 0 202.

Fig . 5

I nstantaneous surface pressure coefficient distr i

Fig . 8 P ar tial view of po ints distr ibution around

butions for NACA 0012 airfoil in the fort h circle

the NACA64A010 airfo il

of motion

Distribut ion of point s around t he NACA64 A010 airf oil is show n in Fig. 8. T here are 270 points dist ributed around t he airfoil and 5681 points distribut ed in the w hole comput at ional f ield. In this case, 120 t ime st eps are g iven for one oscil lation circle during t he computation because t he an g le f requency in here is higher. Inst ant aneous lift coef ficient s C l vs t he instan taneous angle of att ack are plott ed in Fig 9. T he Fig 6

Lift coefficients vs the instantaneous angle of attack

Fig. 9

Lift coefficients vs the instantaneous angle of attack

Fig . 7

3. 3

Mo ment coefficients vs the instantaneous

present results compare w ell wit h the ex periment al dat a[ 10] . T he in phase and in quadrature com po

angle of attack

nents of t he 1st harmonic of unst eady surf ace pres

Unsteady flow around osci llating NACA64 A010 airfoil T o furt her underst and t he propert ies of t he

sure coefficients in t he f orth circle of oscillat ion are shown in Fig 10 and com pared w it h the experi ment al results there.

14

WA NG Gang, SU N Ying dan, YE Zheng yin

CJA

met hod to simulat e unsteady flow. T he present met hod can also be directly ex tended to t wo dimen sional unsteady v iscous f low and three dimensional unst eady f low . References [ 1]

Bart h T J. A gridless Euler/ N avier Stokes solut ion algorit h for com plex aircraft applicat ions [ R ] . AIAA Paper 93 0333, 1993.

[ 2]

M orinishi K . A gridless type solut ion f or high Reynolds num ber multielement f low fields [ R] . A IA A Paper 95 - 1856, 1995.

[ 3]

Liu J L, Su S J . A potentially gridless solut ion met hod for t he compressible Euler/ Navier St okes equations [ R ] . A IAA Paper 96- 0526, 1996.

[ 4]

Onat e E, Idelsohn S. A mesh free finit e point m ethod f or ad vect ive diff usive transport and f luid flow problems [ J] . Com put at ional M echanics, 1998, 21 ( 2) : 283- 292.

[ 5]

Lohner R , S acco C , O nat e E, et al . A finit e point met hod for compressible flow s [ J] . Int ernat ional Journal f or Numeri cal M ethods in Engineering, 2002, 53 ( 8) : 1765- 1779.

[ 6]

Hw ang C J, Y ang S Y . Locally implicit tot al variat ion dim in ishing schemes on mixed quadrilateral t riangular meshes [ J] . A IAA Journal, 1993, 31 ( 11) : 2008- 2015.

Fig . 10

T he in phase and in quadrature co mponents o f

[ 7]

st eady Euler equat ions [ J] . A eronautical Journal, 1994, 98

t he 1st harmonic of unsteady surface pressur e coefficients in the forth cir cle

G ait on de A L. A dual t ime met hod for t he solution of t he un ( 10) : 283- 291.

[ 8]

Lu Z L. G eneration of dynamic grids and computat ion of un st eady t ransonic f low s aroune assemblies [ J] . Chinese Journal

4

Conclusions

In t his paper, a gridless algorithm has been developed t o simulat e unsteady f low . A quadrantal point infilling st rat eg y is employed t o g enerat e t he cloud of points. A point moving algorithm is int ro duced t o ensure the clouds of points follow ing w ith the movement of body boundaries. T he spat ial derivat ives are directly det ermined by using local least squares curve f its in each cloud of point s, and then numerical flux can be obt ained. A dual t ime met hod is used to advance t he flow equations in time. T he computat ion of NACA0012 airfoil in transonic st eady flow proves t he accuracy of spat ial discret ization in t he present gridless method. T he unst eady flow of NACA0012 airfoil and NACA64A010 airfoil oscillat ion in pitch about their quart er chord are simulat ed and the results are in g ood agreement w it h t he ex perimental data. T hese test cases demonst rat e the validit y and pract icalit y of t he present method. T he present research has made prim ary f oundat ion for applying the gridless

of A eronautics, 2001, 14( 1) : 1- 5. [ 9]

Landon R H. N ACA 0012 oscillat ory and transient pit ching

[ 10]

D avis S S. N ACA 64A 010 ( NA SA A mes model) oscillatory

[ R ] . A GA RD R eport 702, AG AR D, 1982. Dataset 3. pit ching[ R] . AG AR D R eport 702, A GA RD , 1982. Dataset 2.

Biographies: WANG Gang Born in 1977 in Hubei province, a g raduate student for the doc toral deg ree in Nor thw ester n Polytechnic U niversity. He is resear ching in compu tational fluid dynamics. T el: 029 88491342, E mail: wang gang @ nwpu. edu. cn SUN Ying dan Bor n in 1979 in L iaon ing pro vince, a post g raduate student in College of Aeronautics of Northwestern P olytechnic U niv ersity. She is w orking on the g ridless method in CFD. T el: 029 88493955, E mail: sunying dan2002@ tom. co m YE Zheng yin Born in 1963 in Hubei prov ince, professor of Northwestern Polytechnic U niv ersity. He is r esearching in CFD and aero elastic dynamics. T el: 029 88491374, E mail: yezy @ nwpu. edu. cn