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 Balancing, transport equations Computer Fluid Dynamics E CFD

Balancing CFD4 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

Transported property  CFD4  Property related to unit mass P=  related to unit volume (  is balanced in the fluid element) P diffusive molecular flux of property  through unit surface. U nits of P multiplied by m/s Constitutive laws and transport coefficients c having the same unit m 2 /s P = - c  P Mass1  0 Momentum Viscous stresses Newton’s law (kinematic viscosity) Total energy Enthalpy EhEh  E  h=  c p T Heat flux Fourier’s law (temperature diffusivity) Mass fraction of a component in mixture AA  A =  A diffusion flux of component A Fick’s law (diffusion coefficient) 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.

Integral balancing - Gauss CFD4 Control volume balance expressed by Gauss theorem accumulation = flux through boundary Variable  can be  Vector (vector of velocity, momentum, heat flux). Surface integral represents flux of vector in the direction of outer normal.  Tensor (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 . Divergence of  projection of  to outer normal  dd

Fluid ELEMENT fixed in space CFD4 x y z xx zz S outh W est T op E N orth B ottom Motion of fluid is described either by  Lagrangian coordinate system (tracking individual particles along streamlines)  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 3D case are usually marked by letters W/E, S/N, and B/T.

Mass balancing (fluid element) CFD4 Accumulation of mass Mass flowrate through sides W and E x xx y z zz S outh W est T op E N orth B ottom xx yy

Mass balancing CFD4 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  r z r  rr zz u v w

Fluid PARTICLE / ELEMENT CFD4 Modigliani 20 km/h running observer Time derivatives -at a fixed place -at a moving coordinate system… 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).Jaumann derivatives

Fluid PARTICLE / ELEMENT CFD4 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

Balancing  in fixed F luid E lement CFD4 [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

Balancing  in fixed F luid E lement CFD4 Accumulation of  inside the fluid element Flowrate of  out of Fluid element Rate of  increase of fluid particle Conservation form (  balance) Nonconservation form

Integral balance  in F luid E lement CFD4

Moving Fluid element CFD4 Fluid element V at time t Fluid element V+dV at time t+dt Amount of  in new FE at t+dt Convection inflow at relative velocity Diffusional inflow of  Integral balance of property  velocity of FE velocity of particle (flow) Terms describing motion of FE are canceled moving control volume

Moving Fluid element CFD4 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…)

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 Internal volumetric sources of  (e.g. gravity, reaction heat, microwave…) Reynolds transport theorem CFD4

Integral/differential form CFD4 Integral form Differential form All integrals can be converted to volume integral s (Gauss theorem again)