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Show (2) - momentum : o _ --p ot o _ u~ + -- p oXa o +--15 oXa - energy of Ua u~ + --ox~ u~ a u ~ = 0 ; ~ o 0 0 ~ (3) -- P h + -- p uah + --P u"a hIt ot oXa oXa with: h = CpT + \ (~a'~a) o 1, 2 wR = term representing radiation source here G r~resen~s the statistical average of G and G = pG/p its FAVRE average The models of turbulence and radiation which are discussed in the following paragraph will provide expressions of : -~~and~" state (4) o - wR To these equations is added the equation of - P = P R T METHOD OF SOLUTION - The numerical technique used to solve the problem is that of D. DUTOYA (1). It has been applied to two and three dimensional flows in combustion chambers (2,3), to diffusers (4), and furnaces (5). The method comprises two parts : - a spacial discretization - an integration over time Spacial discretization - The system formed by the equations (1), (3) can be written: of (5) a (f) ot * f is the "vector" Ng [ p ] PUa h f It may include other components, arising, for example, from the turbulence model (k,E, ... ) * a is an operator which brings into play the spatial derivates up to the order of 2 : it is representative of the convection and diffusion transfer and contains the source laws (~ for example). The object of the operation of spatial discretization is to replace the partial derivates of the operator a by algebraic equations. The technique chosen is that of finite volumes, which has the advantage of ensuring conservativity (1,6). It consists in partitioning the domaine of calculation into Nv cells, each of a volume Vi (i = 1, ••• , Nv), and then writing for each of these cells a record of mass, momentum, energy, • •• In order to calculate the components of momentum, a mesh that is shifted in relation to !hat relating to the scalar quantities (13', ti, k, ••• ) is used. Integration over time - After spatial discretization, we are confronted with a problem of the type : 128 (6) df -- = A (f) dt f is now a vector with NgxNv components. A is a non-linear function of f. The integration method of D. DUTOYA (1) is similar to that established by R.M. BEAM and R.F. WARMING (7). The method is implicit and makes it possible to use values of ~t greater than that set by the CFL (COURANT-FRIEDRICH-LEWY) criterium. Normally the value ~t ~ 50 CFL is chosen. BOUNDARY CONDITIONS - * at the walls : - the components of the velocity and their derivatives in the direction parallel to the tangent to the wall are nil : - the normal derivative of the component parallel to the tangent is calculated using the friction stress. - the normal derivates of density and pressure are nil. * on a symmetrical plane or axis laws of symmetry are applied. * at the entry to the domain : the stagnation pressure (or the flowrate) and the stagnation enthalpy are set. * at the exit : the static pressure is set. MESHING OF DOMAIN OF CALCULATION - The geometry of the system calculated may be either flat two-dimensional or axisymmetric. In the cells (N+2) (M+2), NxM are within the physical domain and 2(M+N+2), located at the edge of the domain, are intended for processing of limit conditions. PHYSICAL MODELS AND COMPUTER PROGRAM MODELS OF TURBULENCE - Expression must be obtained for turbulent diffusion flows of momentum u"au"~, and heat u"ah". A hypothesis of motion by gradient is used : (7) - ~ = IJ.t is o~a o~~ --+--- ox~ oXa 2 IJ.t OUy -3--- -- 6a~ is OXy dynamic eddy viscosity KRONECKER delta 2 ,... - -3- k6a~ kinetic energy of the turbulence per unit mass (8) P rt Prt ~ 1 _q=~ oh PPrt oXa turbulent Prandtl number In order to evaluate IJ.t, two solutions are possible : a) IJ.t : constant IJ.t must be linked to a characteristic flow length : (9) IJ.t = CpU L C is an adjustable constant. |