Boundary Domain Integral Method for the Space Time Dependent Viscous Incompressible Flow

I. ZAGAR, P. SKERGET, A. ALUJEVIC

Faculty of Engineering, University of Maribor, Slovenia

Summary

The partial differential set of equations, governing the laminar or the turbulent motion of viscous incompressible fluid is known as nonlinear Navier-Stokes equations. They constitute the statement of the basic conservation balance of mass, momentum and energy applied to a control volume, i.e. the eulerian description. The time dependent set of governing equations is handled since it is more stable, and what turns to be very important - the time dependent approach does not presume the existence of steady state solution which may not even exist. Various approaches exist for the turbulent flow prediction, i.e. full turbulence simula­tion, large eddy simulation, Reynolds averaged models, etc. The averaged form of the Navier-Stokes equations throught the Reynolds decompositIOn of instantaneous value of each variable into a time averaged mean value and an instantaneous fluctuation is still the most commonly used approach in the numerical simulation of the turbulence. The buoyancy force can play an important role in a nonisothermal flow, especially in the case of mixed or pure natural convection. The Boussinesq approximation is used to introduce the buoyancy force. Boundary-domain integral approach offers some important features. Due to the fun­damental solutions, a part of the transport mechanism is transferred to the boundary, producing a very stable and accurate numerical scheme. Diffusion transport part and potential part of the flow is described only by boundary integrals while the convection is described by domain integrals. Boundary vorticity values are included in integral form in kinematic boundary integral formulation, so there is no need to use approximate for­mulae to determine boundary vorticity conditions. One has to solve implicit systems of equations only for the boundary values, while all internal domain values are computed in an explicit manner. Existing limitations of boundary-domain integral method in fluid dynamics concerning the computational cost and computer memory demands (matrices are fully populated and non symetric, expensive evaluation of the integrals) can be reduced by using sub­domain technique, efficient block solver and clustering. Stability of the method at higher Reynolds numbers can be improved with introducing a part of the convection in to the system matrix.

L. Morino et al. (eds.), Boundary Integral Methods© Springer-Verlag Berlin, Heidelberg 1991

511

GOVERNING EQUATIONS

The partial differential equations set, governing the motion of viscous incompressible

fluid is known as nonlinear Navier-Stokes equations expressing the basic conservation

balance of mass, momentum, and energy.

avo -' =0 aXi

(1)

av; aViv; 1 ap a2v; -+--= ---+vo---g;F (2) at ax; Po ax; aXiax;

Introducing the vorticity vector W; and the vector potential W; of the solenoidal velocity

field, the computation of the flow is divided into the kinematics given by the vector

Poisson's elliptic equation

a2W; -a a +w;=O Xi Xi

and into the kinetics described by the vorticity transport equation

(3)

(4)

The buoyancy effect is included by Boussinesq approximation and the energy equation

is given as follows

(5)

The following linear relation between the fluid density and temperature is usually used

P - Po = F = -(3 (T - To) (6) Po

In the region of density anomaly, for example in water, the nonlinear term must be

taken into account

P - Po = F = (0.066576 T - 0.008322 T2) / Po Po

(7)

Direct simulation of turbulence is the solution of the Navier-Stokes equations for the

complete details of the turbulent flow. Such simulation is necessarily t'hree-dimensional

and time dependent and is too expensive to be widely used. The averaged form of

512

the Navier-Stokes equations is still the most commonly used approach in the practical

calculation of the turbulent flows.

When considering the turbulent flow a time averaged form of the Navier-Stokes equa­

tions is usually employed through the Reynolds decomposition of the instantaneous

value of each variable, e.g. velocity Vi, into a time-averaged mean value Vi and an

instantaneous deviation or fluctuation v;' from the mean value (Vi = Vi + v;'). The stress tensor Tij can be written as a sum of the viscous part written for the time

mean values and the Reynolds turbulent stresses (-pv/v/) in the form

_ ( 8Vi iJtJj) -.-. Tij = -p6ij + 170 -8 + -8 - PVi Vj Xj Xi

(8)

The Reynolds mean momentum equations are given as follows

(9)

(10)

(11)

With the exception of the additional Reynolds stress term, the mean velocity of a

turbulent flow and the instantaneous velocity of a laminar flow satisfy the same set of

differential equations.

The mean flow vorticity transport equation can be derived similary by decomposing

the vorticity vector into a time averaged mean value Wi and an instantaneous deviation

W;' from the mean value Wi = Wi + w/ and by taking the time averaged form of the

instantaneous vorticity equation [6]

8Wi 8Vj Wi 8Wj Vi 82wi 8F 8v/w/ 8w/v/ - + -- = -- + V o--- - fijkgk- - -- + --- (12) 8t 8xj 8xj 8xj8xj 8xj 8xj 8xj

The turbulent stress terms (-pv/ v/) are usually interpreted in the Boussinesq manner

similarly to the viscous stress terms

-, -, (8Vi 8vj) 2 - p V· V· = 17t - + - - -6·p k o , 1 8Xj 8Xi 3 '1 0

(13)

in which 17t is turbulent or eddy dynamic viscosity and k mean turbulence kinetic energy.

When the temperature is regarded as a passive scalar, the term (-T'v/) is assumed to

be related to the mean temperature as follows [3]

513

-,-, aT -Tv· =at-

1 aXi (14)

now at being the eddy temperature diffusivity.

BOUNDARY-DOMAIN INTEGRAL EQUATIONS

Nonlinear Diffusion-Convective Equation

The mean diffusion-convective transport equations for the momentum, vorticity, tem­

perature, turbulent kinetic energy, dissipation, etc. can be recognized to be formally of

the same type as a nonhomogeneous nonlinear parabolic partial differential equation of

the form

(15)

The nonhomogeneous term b represents pseudo-body forces expressing the convection

and eddy diffusion term.

Using Green's theorem for the scalar functions and the parabolic fundamental solution

u*, one can derive the following boundary-domain integral statement to eq.(15) in the

incremental form for the time step T = tF - tF- 1 equating b = -aa (ataaU - viu) x] 'Xl

({tp au' li tP au c(~)u(~,tF) + ao iT it u-a dt dr = ae-u* dt dr

r tp-l n r tp-l an

- { t P U vnu* dt dr - { {tp (at aau - UVi) au* dt dO ir itP_l in i tP_1 Xi aXi

+ In UF-l U*F-l dO (16)

when ae is the effective diffusivity (ae = ao + at).

Velocity-Vorticity Formulation

The boundary domain integral statement for the flow kinematics can be derived from

the vector elliptic eq.(3) applying Green's theorem for the vector functions and the

elliptic fundamental solution U *, resulting in the following statement written in the

vector notation [4,71

514

The integral statement (17) represents three scalar equations and only two of them are

independent. It is completly equivalent to the continuity equation and vorticity defini­

tion, expressing the kinematics of the laminar and turbulent incompressible flow in the

integral form. Boundary velocity conditions are included in boundary integrals, while

the domain integral gives the contribution of the vorticity field to the development of

the velocity field. Equation enables the explicit computation of the velocity vector in

the interior of the domain (c(~) = 1). When the unknowns are the boundary vortic­

ity values or the tangential velocity component to the boundary, one has to use the

tangential component of the vector eq.(17) [8]

cwTt(~)xtt(~) + Tt(~)x 1r(~ u* . Tt)tt df Tt(~)x 1r(~ u*xTt)xtt df

+ TtWxfoW'xVu*dll (18)

or when the normal velocity to the boundary is unknown, the normal component of the

vector eq.(17) has to be used

c(~) Tt( ~). It(~) + TtW·1r (V u* . Tt)1t df TtW·1r (V u*xTt)xlt df

+ Tt(~)·fo W'xVu*dll (19)

in order to perform appropriate implicit system of discretized equations. Unknown

boundary vorticity values are expressed in the integral form eq.(18) within the domain

integral, excluding a need to use approximate formulae for determining boundary vor­

ticity values, which would bring some additional error into the numerical scheme.

Describing the laminar transport of the vorticity and temperature in the integral

statement, one has to take into account that each instantaneous component of the

vorticity vector and temperature obey a nonhomogenous parabolic equation [5]; the fol­

lowing boundary-domain integral formulations can be derived for the vorticity transfer

and temperature transport

l1tP au' l1tP aw· C(~)Wi(~,tF) + Va W'-a dt df = Va -'u* dt df r tp-1 n r tp-1 an

(20)

515

i l tF [ ltF au: [ - T v,.u· dt dr - TV;-a + TF - 1 U'F-l dO r tF-l 0 tF-l X; 0

(21)

Turbulent transport equations for the time mean values of the vorticity and tem­

perature eq.(12) and eq.(ll) are formally identical to the nonlinear diffusion-convective

eq.(15). Following the same idea the integral statements can be written according to

eq.(16) for the time mean vorticity and temperature.

(22)

_ r ltF Tv,.u· dt dr _ r ltF (at aT _ Tv;) au' dt dO Jr tF_l Jo tF-l ax; ax;

+ In TF-l U'F-l dO (23)

SUB-DOMAIN TECHNIQUE

In general sub-domain technique is used to model various material properties of piece­

wise homogeneous zones, in order to overcome geometry caused problems, and to sig­

nificantly reduce the computer time and memory demands, what is specially true in

the fluid flow computation. The numerical procedure presented in this paper, can be

applied to each of the subregions as they are separated from the each other. The final

implicit system of equations for the external boundary and interface boundaries of the

whole region is obtained by adding the set of equations for each subregion together

considering compatibility and equilibrium conditions between interfaces.

516

Let us now first consider the kinetics of the flow governed generaly by the eq.(20).

The compatibility and equilibrium conditions to be applied at the interface r I between

0 1 and 0 2 are respectively

{J}} = {J}i, (24)

In the kinematics the only proper compatibility conditions are applied to the tan­

gential and normal velocity component

(25)

On the internal interfaces the boundary vorticity and boundary vorticity flux values

are determined in the kinetics. In the kinematics the only unknowns on the interface

are tangential and normal velocity components and for the tangential (18) or normal

(19) form of the kinematic equation has to be employed. In general, for all source points

lieing on the external or interface boundaries the tangential or normal component of

the kinematic statement has to be used, while component kinematic eq.(17) has to be

employed to evaluate explicitely unknown values in the interior of the each subdomain.

TEST EXAMPLES

Two examples are given here in order to demonstrate the applicability of the proposed

method for laminar fluid flow.

Free convection in cylinders

z

Fig. 1: Geometry of the cylinder

- -- - - - - - .. /' - -- - - - -- , ,/ -- ---.... ----

......... ...... " \ / / ........ \' ......... ...... " \ \ \ " ~ \ \ \ I

'\. I ) , I I , ...... ./ / / / / I -- .-/ .,/ /' ,.- / I - - - - - - ." ,

---./ ...... /

( - " - \ \ - ) " -...... /

./

---

Fig. 2: Velocity fields and isotherm contours in inclined cylinder

"I = 1350 , 1> = 00 for Ra = 6250 at t = 8 and 40 s

517

518

The free convection in inclined cylinders has been studied first. The BEM results are

compared with FDM results, Bontoux [1].

The cylinder geometry is depicted in Fig. 1, where R is radius, L lenght and A = L/ R

aspect ratio. The inclination angle I is referred to the vertical axis. The two circular

endwal!s are kept at constant temperatures Th = 1 and Tc = 0, while the side wall is

assumed perfectly conducting.

The linear 3-node triangular and 4-node quadrilateral boundary elements were used

to model the boundary, while linear 8-node and 6-node brick internal cells are applied

to discretise the domain. Mesh sizes M = 5 x 16 x 9 in radial, azimuthal and axial

directions were used. The time step Dot = tF - t F - 1 = 1s, and the under-relaxation

factor 0.1 were used. Free-convection motion numerical results for the A = 5 cylinder

when the axis is inclined at an angle I = 1350 with the gravity vector are presented

for the Rayleigh number Ra = 6250. Velocity fields and isotherm contours for inclined

cylinder are depicted in Fig. 2.

Square cavity with water

A closed cavity with natural convection in water due to temperature difference from the

left (8 0 e) to right side (00 e), while top and bottom walls are kept to be adiabatic,

has been studied for Rayleigh number value of 105 • A mesh of 40 elements (80 nodes)

and 400 internal cells has been used .

\ .. _----, .... ----_ .... ,

Fig. 3: Velocity and temperature field distribution for steady state (t = 140 s)

Figure 3 gives the final steady state (t = 140 s) of both velocity and temperature

distributions. There are two separated circural zones observerd, while in the middle

(T = 40 e) a symmetry line is developed.

519

References

[I] Bontoux,P., Smutek,C., Roux,B., Extremet,G.P., Schiroky,G.H., Hurford,A.C.,

Rosenberg,F.: Finite Difference Solutions for Three Dimensional Buoyancy Driven

Flows in Inclined Cylinder, Vo1.3, 3rd Int. Conf. on Num. Meth. for Nonlinear Prob­

lems, Dubrovnik. Pineridge Press, 1986.

[2] Brebbia,C.A., Telles,J.F.C., Wrobel,L.C.:Boundary Element Methods-Theory and

Applications, Springer-Verlag, New York, 1984.

[3] Nagano,Y., Kim,C.:A Two-Equation Model for Heat Transport In Wall Turbulent

Shear Flows, Journal of Heat Transfer, Vol.ll0, 1988.

[4] Skerget,P., Alujevic,A., Zagar,!., Brebbia,C.A., Kuhn,G.:Time Dependent Three Di­

mensional Laminar Isochoric Viscous Fluid Flow by BEM, 10th Int. Conf. on BEM,

Southampton, Springer-Verlag, Berlin, 1988.

[5] Skerget,P., Alujevic,A., Brebbia,C.A., Kuhn,G.:Natural and Forced Convection Sim­

ulation Using the Velocity- Vorticity Approach, Topics in Boundary Element Research

(Ed. by Brebbia C.A.), Vo1.5, Ch.4, Springer-Verlag, Berlin, 1989.

[6] Tennekes,H., Lumly,J.L.:A first Course in Turbulence, The MIT Press, Boston, 1972.

[7] WU,J.C., Guicat,U., Wang,C.M., Sankar,N.L.:A Generalized Formulation for Un­

steady Viscous Flow Problems, Topics in Boundary Element Research (ed. Brebbia

C.A.), Vo1.5, Ch.3, Springer-Verlag, Berlin, 1989.

[8] Zagar,!., Skerget,P.: Boundary Elements for Time Dependent 3-D Laminar Viscous

Fluid Flow, Mechanical Engineering Journal, Vol. 10-12, Ljubljana, 1989.