058:0160 Jianming Yang Chapter 5 10 Fall 2012 3 Common Dimensionless Parameters for Fluid Flow Problems Most common physical quantities of importance in fluid flows are: (without heat transfer) 1 2 3 4 5 6 7 8 V, , g, , , K, p, L velocity density gravity viscosity surface tension compressibility pressure change length n=8 m=3 5 dimensionless parameters VL inertia forces V 2 / L 1) Reynolds number = viscous forces V / L2 R e Rcrit distinguishes among flow regions: laminar or turbulent value varies depending upon flow situation V inertia forces 2) Froude number = gL gravity force V 2 / L Fr important parameter in free-surface flows V 2 L inertia force V 2 / L 3) Weber number = surface tension force / L2 We important parameter at gas-liquid or liquid-liquid interfaces and when these surfaces are in contact with a boundary 058:0160 Jianming Yang Chapter 5 11 Fall 2012 4) Mach number = V V inertia force compressib ility force k / a Ma where a is the speed of sound in fluid Paramount importance in high speed flow (V > c) 5) Pressure Coefficient = p V 2 pressure force p / L inertia force V 2 / L Cp (Euler Number) 4 Nondimensionalization of the Basic Equation It is very useful and instructive to nondimensionalize the basic equations and boundary conditions. Consider the situation for and constant and for flow with a free surface Continuity: V 0 Momentum: DV p gz 2V Dt 058:0160 Jianming Yang Chapter 5 12 Fall 2012 Boundary Conditions: 1) fixed solid surface: V 0 2) inlet or outlet: V = Vo p = po 3) free surface: w (z = ) t p pa Rx1 Ry1 All variables are now nondimensionalized in terms of and U = reference velocity L = reference length V* V U t* tU L x* x L p* p gz U 2 All equations can be put in nondimensional form by making the substitution V V* U t * U t t * t L t * 058:0160 Jianming Yang Chapter 5 13 Fall 2012 î ĵ k̂ x y z x * y* z * * î * ĵ * k̂ x x y y z z 1 * L u 1 U u * * and etc. Uu x L x * L x * Result: * V* 0 DV * *p* *2 V* Dt VL 1) 2) 3) V* 0 V V* o U * w * t * Re-1 p* po V 2 p* po gL * *1 *1 z R R x y 2 2 2 V U V L pressure coefficient Fr-2 We-1 058:0160 Jianming Yang Fall 2012 Chapter 5 14