Momentum Transport Force

This scheme is only used with the ALE formulation (Arbitrary Lagrangian Eulerian) and in the CFD version of Radioss. The force is calculated using the relation:
図 1.
F t r m i I = ( 1 + η I ) ρ Φ I ( w j − v j )   ∂ v i ∂ X j   V MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaake aadaahbaWcbeqaaiaadshacaWGYbGaamyBaaaakiaadAeadaqhaaWc baGaamyAaaqaaiaadMeaaaGccqGH9aqpdaqadaqaaiaaigdacqGHRa WkcqaH3oaAdaWgaaWcbaGaamysaaqabaaakiaawIcacaGLPaaacqaH bpGCcqqHMoGrdaWgaaWcbaGaamysaaqabaGcdaqadaqaaiaadEhada WgaaWcbaGaamOAaaqabaGccqGHsislcaWG2bWaaSbaaSqaaiaadQga aeqaaaGccaGLOaGaayzkaaGaaGjbVpaalaaabaGaeyOaIyRaamODam aaBaaaleaacaWGPbaabeaaaOqaaiabgkGi2kaadIfadaWgaaWcbaGa amOAaaqabaaaaOGaaGPaVlaadAfaaaa@5B70@
Where,
w MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4Daaaa@36F2@
Grid velocity
ν
Material velocity
V MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4Daaaa@36F2@
Element volume
η
Upwind coefficient (user-defined, default = 1 for full upwind)

When a Lagrangian formulation is used, the values of w j MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4DamaaBa aaleaacaWGQbaabeaaaaa@380D@ and ν j MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyVd42aaS baaSqaaiaadQgaaeqaaaaa@38C9@ are equal. Thus, 式 1 is equal to zero.

Upwinding Technique

An upwinding technique is introduced to add numerical diffusion to the scheme; otherwise it is generally under diffusive and thus unstable. The upwind coefficient used in 式 1 is calculated by:
図 2.
η I = η s i g n ( ∂ Φ I ∂ X j ( v j − w j ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaake aacqaH3oaAdaWgaaWcbaGaamysaaqabaGccqGH9aqpcqaH3oaAcaWG ZbGaamyAaiaadEgacaWGUbWaaeWaaeaadaWcaaqaaiabgkGi2kabfA 6agnaaBaaaleaacaWGjbaabeaaaOqaaiabgkGi2kaadIfadaWgaaWc baGaamOAaaqabaaaaOWaaeWaaeaacaWG2bWaaSbaaSqaaiaadQgaae qaaOGaeyOeI0Iaam4DamaaBaaaleaacaWGQbaabeaaaOGaayjkaiaa wMcaaaGaayjkaiaawMcaaaaa@519A@
Development of a less diffusive flux calculation is currently under investigation.
図 3.
F i I = σ i j ∫ V ∂ Φ I ∂ X j d V MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaake aacaWGgbWaa0baaSqaaiaadMgaaeaacaWGjbaaaOGaeyypa0Jaeq4W dm3aaSbaaSqaaiaadMgacaWGQbaabeaakmaapefabaWaaSaaaeaacq GHciITcqqHMoGrdaWgaaWcbaGaamysaaqabaaakeaacqGHciITcaWG ybWaaSbaaSqaaiaadQgaaeqaaaaakiaadsgacaWGwbaaleaacaWGwb aabeqdcqGHRiI8aaaa@4CB6@

This option is activated with the flag INTEG (only in the CFD version).