$$$$ CLMI NOTICE CHAT 11/09/12 21:15:29 7124 DATE 11/09/12 Operateur CLMI Voir aussi : SYNTAXE : ------- Syntaxe EQEX: ... 'EQEX' ... 'OPER' 'CLMI' ferm equa 'UE' 'DUE' 'I1NM' 'I2NM' 'INCO' 'I1N' (... 'EQEX' ... 'OPER' 'CLMI' ferm equa 'UE' 'DUE' 'I2NM' 'I1NM' 'INCO' 'I2N') DESCRIPTION : ------------- The CLMI operator discretises the differential equations that apply in the boundary layers: these equations are the integral form of the momentum equation(1), the energy equation(2) and the displacement equation(3): d(D2) H+2 d(Ue) Cf (1) ----- + ---- ----- D2 = ---- dX Ue dX 2 d(D3) 3 d(Ue) (2) ----- + --- ----- D3 = 2Cd dX Ue dX d(D-D1) 1 d(Ue) (3) ------- + --- ----- (D-D1) = Ce dX Ue dX D2: Momentum thickness D3: Energy thickness D1: Displacement thickness D: Boundary layer thickness The CLMI operator discretises only one equation, so that in a case of a 2 equations method, the operator must be called two times. Turbulent and laminar boundary layers can be calculated with the CLMI operator. Several resolution methods are available. 1/Laminar boundary layer: _________________________ a/Blasius approximation: ---------------------- This method can be applied in order to calculate boundary layers on flat plates with very low pressure gradient. b/Approximate method due to Von Karman-Polhausen: ----------------------------------------------- Boundary layers with pressure gradient can be calculated by using this method. Nevertheless, the pressure gradients must vary slowly. c/2-equation method: ------------------ It is the most general method, it can be used for boundary layers with any pressure gradient. 2/The turbulent boundary layer: _______________________________ a/Head's method: -------------- This method is based on Head 's relations. It provides good results. b/Michel's method: ---------------- Generally, it provides better results than the previous method. However, it leads to a separation shape factor lower than the one obtained by experiments and by the previous method, so with this method the boundary layer can separate earlier than in the reality. Comments ________ ferm FLOTTANT Method used for the computation of the boundary layer 1 = Laminar boundary layer, Blasius approximation 2 = Laminar boundary layer, method due to Von Karman-Pohlausen 3 = Laminar boundary layer, 2-equation method 4 = Turbulent boundary layer, Michel's method 5 = Turbulent boundary layer, Head's method equa FLOTTANT Equation discretised by the CLMI operator 1 = Momentum equation 2 = Energy equation 3 = Displacement equation UE CHPOINT External velocity distribution DUE CHPOINT Gradient of UE (In the present version, the CLMI operator can not calculate the gradient of UE, so DUE must be provided as a argument. In a future version, DUE will be calculated in CLMI). I1NM CHPOINT Unknown of the equation at previous time This unknown is a thickness (unit: m) I2NM CHPOINT Unknown in the second equation at previous time (if 2-equation method) This unknown is a thickness (unit: m) I1N CHPOINT Unknown of the equation This unknown is a thickness (unit: m) Options : (EQEX) _________ The algorithm used is an implicit algorithm. The integral equations are discretised by a SUPG finite element method. Results : _________ In addition to the unknowns I1N, I1NM (I2N, I2NM if 2-equation method), the CLMI operator provides, in the array of unknowns, the coefficient of local skin friction (CF), the shape factor (H). These terms are calculated in CLMI, these are CHPOINT. Remarks : ________ In 1-equation methods, the CHPOINT D1NM is not used, however this argument is necessary, so a CHPOINT must be provided, even if it is not used. In 2-equation methods, take preferably the momentum equation as the first equation, so I1N and I1NM correspond to the momentum thickness (D3) at time t and t-dt. The second equation is the energy equation (I2N, I2NM are energy thicknesses(D3)) or the displacement equation (I2N, I2NM correspond to the difference between the boundary layer thickness and the displacement thickness).
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