CFD 4 Computer Fluid Dynamics 2181106 E 181107

CFD 4 Computer Fluid Dynamics 2181106 E 181107 Balancing, transport equations Remark: foils with „black background“ could be skipped, they are aimed to the more advanced courses Rudolf Žitný, Ústav procesní a zpracovatelské techniky ČVUT FS 2013

CFD 4 Balancing CFD is based upon conservation laws -conservation of mass -conservation of momentum m. du/dt=F (second Newton’s law) -conservation of energy dq=du+pdv (first law of thermodynamics) System is considered as continuum and described by macroscopic variables

CFD 4 Transported property This table presents nomenclature of transported properties for specific cases of mass, momentum, energy and component transport. Similarity of constitutive equations (Newton, Fourier, Fick) is basis for unified formulation of transport equations. Mass Property P= P diffusive related to unit mass related to unit volume ( is balanced in the fluid element) molecular flux of property through unit surface. Units of 0 1 P multiplied by m/s Constitutive laws and transport coefficients c having the same unit m 2/s P= - c P Newton’s law (kinematic viscosity) Momentum Viscous stresses Total energy E Enthalpy h E h= cp. T Mass fraction A A= A of a component in mixture Fourier’s law (temperature diffusivity) Heat flux Fick’s law (diffusion coefficient) diffusion flux of component A

CFD 4 Integral balancing - Gauss Control volume balance expressed by Gauss theorem accumulation = flux through boundary Divergence of projection of to outer normal Variable can be ØVector (vector of velocity, momentum, heat flux). Surface integral represents flux of vector in the direction of outer normal. ØTensor (tensor of stresses). In this case the Gauss theorem represents the balance between inner stresses and outer forces acting upon the surface, in view of the fact that is the vector of forces acting on the oriented surface d. d

CFD 4 Fluid ELEMENT fixed in space Motion of fluid is described either by Ø Lagrangian coordinate system (tracking individual particles along streamlines) Ø Eulerian coordinate system (fixed in space, flow is characterized by velocity field) Balances in Eulerian description are based upon identification of fluxes through sides of a box (FLUID ELEMENT) fixed in space. Sides if the box in the 3 D y case are usually marked by letters W/E, S/N, and B/T. z Top North E West South Bottom z x x

CFD 4 Mass balancing (fluid element) Accumulation of mass Mass flowrate through sides W and E z y North y West Top E South x Bottom x z x

CFD 4 Mass balancing Continuity equation written in index notation Continuity equation written in symbolic form (the so called conservative form) Symbolic notation is independent of coordinate system. For example in the cylindrical coordinate system (r, , z) this equation looks different w z z r r r v u

CFD 4 Fluid PARTICLE / ELEMENT Time derivatives -at a fixed place -at a moving coordinate system… running observer 20 km/h In other words: Different time derivatives distinguish between time changes seen by an observer that is steady ( / t), an observer moving at a prescribed velocity (d /dt), observer translated with the fluid particle (D /Dt - material derivative) or moving and rotating with the fluid particle ( / t - Jaumann derivatives). Modigliani

CFD 4 Fluid PARTICLE / ELEMENT Fluid element – a control volume fixed in space (filled by fluid). Balancing using fluid elements results to the conservative formulation, preferred in the CFD of compressible fluids Fluid particle – group of molecules at a point, characterized by property (related to unit mass). Balancing using fluid particles results to the nonconservation form. Rate of change of property (t, x, y, z) during the fluid particle motion Material derivative Projection of gradient to the flow direction

CFD 4 Balancing in fixed Fluid Element [Accumulation in FE ] + [Outflow of from FE by convection] = intensity of inner sources or diffusional fluxes across the fluid element boundary This follows from the mass balance These terms are cancelled

CFD 4 Balancing in fixed Fluid Element Conservation form ( balance) Nonconservation form Rate of increase of fluid particle Flowrate of out of Fluid element Accumulation of inside the fluid element

CFD 4 Integral balance in Fluid Element

CFD 4 Moving Fluid element moving control volume velocity of particle (flow) Fluid element V+d. V at time t+dt velocity of FE Integral balance of property Fluid element V at time t Amount of in new FE at t+dt Convection inflow at relative velocity Diffusional inflow of Terms describing motion of FE are canceled

CFD 4 Moving Fluid element You can imagine that the FE moves with fluid particles, with the same velocity, that it expands or contracts according to changing density (therefore FE represents a moving cloud of fluid particle), however the same resulting integral balance is obtained as for the case of the fixed FE in space: Diffusive flux of superposed to the fluid velocity u Internal volumetric sources of (e. g. gravity, reaction heat, microwave…)

CFD 4 Moving Fluid element (Reynolds theorem) You can imagine that the control volume moves with fluid particles, with the same velocity, that it expands or contracts according to the changing density (therefore it represents a moving cloud of fluid particles), however: The same resulting integral balance is obtained in a moving element as for the case of the fixed FE in space Diffusive flux of superposed to the fluid velocity u Reynolds transport theorem Internal volumetric sources of (e. g. gravity, reaction heat, microwave…)

CFD 4 Integral/differential form All integrals can be converted to volume integral s (Gauss theorem again) Integral form Differential form