Motivation & model Scheme Results Temperature dependent material properties in incompressible flow Jan Pech Institute of Thermomechanics of the Czech Academy of Sciences jpech@it.cas.cz 11.06.2019 Jan Pech (Czech Acad Sci, Inst Thermomech) Temperature dependent flow 11.06.2019 1 / 18
Motivation & model Scheme Results Institute of Thermomechanics of the Czech Academy of Sciences (www.it.cas.cz) Department of Fluid Dynamics Laboratories of Turbulent Shear Flows transition to turbulence, structure and development of turbulent flow Internal Flows compressible transonic flow (compressor/turbine blades), vibrating profiles, ejectors, ... Environmental Aerodynamics atmospheric boundary layer, atmospheric flows, pollutant dispersion Computational Fluid Dynamics Jan Pech (Czech Acad Sci, Inst Thermomech) Temperature dependent flow 11.06.2019 2 / 18
Motivation & model Scheme Results Simulation of the flow around probes for hot wire anemometry. Thin (cca 5 µ m ) heated wire-low Reynolds number ( Re < 160). Stationary or periodic vortex shedding regime. Data from experiments available (setup for parallel vortex shedding-2D simulation should suffice). 1.8 1.6 normalized value 1.4 ρ */ ρ * 0 1.2 µ */ µ * 0 1 κ */ κ * 0 c* p /c* 0.8 p0 0.6 0.4 1 1.2 1.4 1.6 1.8 2 T*/T* 0 Variables with physical units have ∗ in superscript, subscript 0 denotes values belonging to a chosen reference temperature T ∗ 0 (here T ∗ = 285 K ; 0 ρ ∗ -density; ρ ∗ 0 = ρ ∗ ( T ∗ 0 ); µ ∗ - dynamic viscosity; κ ∗ - thermal conductivity; c ∗ p - spec. heat at constant pressure, variation negligible in this case) [1] Wang A.-B., Tr´ avn´ ıˇ cek Z., Chia K.-C.: On the relationship of effective Reynolds number and Strouhal number for the laminar vortex shedding of a heated circular cylinder, Phys Fluids (2000) 12:6, 1401-1410 [2] Tr´ avn´ ıˇ cek Z., Wang A.-B., Tu W.-Y.: Laminar vortex shedding behind a cooled circular cylinder, Exp Fluids (2014) 55:1679 [3] Marˇ s´ ık F., Tr´ avn´ ıˇ cek Z., Yen, R.-H., et. al.: Sr-Re-Pr relationship for a heated/cooled cylinder in laminar cross flow. Proceedings of CHT-08 ICHMT International Symposium on Advances in Computational Heat Transfer,2008, Marrakech, Morocco Jan Pech (Czech Acad Sci, Inst Thermomech) Temperature dependent flow 11.06.2019 3 / 18
Motivation & model Scheme Results Coupled evolutionary Incompressible Navier-Stokes-Fourier system � ∂ v � = −∇ π + 1 ρ ∂ t + v · ∇ v Re ∇ · [2 µ D + λ ( ∇ · v ) I ] + f v (1a) Jan Pech (Czech Acad Sci, Inst Thermomech) Temperature dependent flow 11.06.2019 4 / 18
Motivation & model Scheme Results Coupled evolutionary Incompressible Navier-Stokes-Fourier system � ∂ v � = −∇ π + 1 ρ ∂ t + v · ∇ v Re ∇ · [2 µ D + λ ( ∇ · v ) I ] + f v (1a) ∂ρ ∂ t + ∇ · ( ρ v ) = 0 (1b) Jan Pech (Czech Acad Sci, Inst Thermomech) Temperature dependent flow 11.06.2019 4 / 18
Motivation & model Scheme Results Coupled evolutionary Incompressible Navier-Stokes-Fourier system � ∂ v � = −∇ π + 1 ρ ∂ t + v · ∇ v Re ∇ · [2 µ D + λ ( ∇ · v ) I ] + f v (1a) ∂ρ ∂ t + ∇ · ( ρ v ) = 0 (1b) � ∂ T � 1 ρ c p ∂ t + v · ∇ T = RePr ∇ · κ ∇ T + f T (1c) Jan Pech (Czech Acad Sci, Inst Thermomech) Temperature dependent flow 11.06.2019 4 / 18
Motivation & model Scheme Results Coupled evolutionary Incompressible Navier-Stokes-Fourier system � ∂ v � = −∇ π + 1 ρ ∂ t + v · ∇ v Re ∇ · [2 µ D + λ ( ∇ · v ) I ] + f v (1a) ∂ρ ∂ t + ∇ · ( ρ v ) = 0 (1b) � ∂ T � 1 ρ c p ∂ t + v · ∇ T = RePr ∇ · κ ∇ T + f T (1c) ..dimensionless formulation in primitive variables, non-conservative form v = ( v 1 , v 2 ) T velocity ( · ) T , ∇ , ∇· matrix vector transposition, gradient, divergence � ∇ v + ( ∇ v ) T � 1 D , I , unit tensor 2 t time T ∗− T ∗ T ∗ T temperature ( T = ∞ or T = ∞ ), T ∗ W temperature on wall T ∗ T ∗ W − T ∗ ∞ ρ = ρ ( T ), ρ � = ρ ( π ) density µ = µ ( T ) dynamic viscosity κ = κ ( T ) thermal conductivity c p = 1 spec. heat capacity at const. pressure, calorically perfect fluid volumetric momentum source, e.g. buoyancy f v heat source, e.g. viscous heating f T Re = L ∗ | v ∗ ∞ | ρ ∗ Reynolds number ( L ∗ is characteristic length, e.g. cyl. diameter) ∞ µ ∗ ∞ cp ∗ ∞ µ ∗ Pr = ∞ Prandtl number κ ∗ ∞ Jan Pech (Czech Acad Sci, Inst Thermomech) Temperature dependent flow 11.06.2019 4 / 18
Motivation & model Scheme Results Coupled evolutionary Incompressible Navier-Stokes-Fourier system � ∂ v � = −∇ π + 1 ρ ∂ t + v · ∇ v Re ∇ · [2 µ D + λ ( ∇ · v ) I ] + f v (1a) ∂ρ ∂ t + ∇ · ( ρ v ) = 0 (1b) � ∂ T � 1 ρ c p ∂ t + v · ∇ T = RePr ∇ · κ ∇ T + f T (1c) ..dimensionless formulation in primitive variables, non-conservative form v = ( v 1 , v 2 ) T velocity The stress tensor represents generalized model of Newtonian fluid, but ( · ) T , ∇ , ∇· matrix vector transposition, gradient, divergence � ∇ v + ( ∇ v ) T � what is λ and π ? 1 D , I , unit tensor 2 t time T ∗− T ∗ T ∗ T temperature ( T = ∞ or T = ∞ ), T ∗ W temperature on wall T ∗ T ∗ W − T ∗ ∞ ρ = ρ ( T ), ρ � = ρ ( π ) density µ = µ ( T ) dynamic viscosity κ = κ ( T ) thermal conductivity c p = 1 spec. heat capacity at const. pressure, calorically perfect fluid volumetric momentum source, e.g. buoyancy f v heat source, e.g. viscous heating f T Re = L ∗ | v ∗ ∞ | ρ ∗ Reynolds number ( L ∗ is characteristic length, e.g. cyl. diameter) ∞ µ ∗ ∞ cp ∗ ∞ µ ∗ Pr = ∞ Prandtl number κ ∗ ∞ Jan Pech (Czech Acad Sci, Inst Thermomech) Temperature dependent flow 11.06.2019 4 / 18
Motivation & model Scheme Results Coupled evolutionary Incompressible Navier-Stokes-Fourier system � ∂ v � = −∇ π + 1 ρ ∂ t + v · ∇ v Re ∇ · [2 µ D + λ ( ∇ · v ) I ] + f v (1a) Physical meaning of ”pressure” in considered model ∂ρ ∂ t + ∇ · ( ρ v ) = 0 (1b) We call thermodynamic pressure the variable acting in the equation of state, e.g. π = ρ RT for ideal gas. But, instead of (1a), we are going to solve � ∂ T � 1 ρ c p ∂ t + v · ∇ T = RePr ∇ · κ ∇ T + f T (1c) � ∂ v � = −∇ p + 1 � 2 µ D − 2 � ρ ∂ t + v · ∇ v Re ∇ · 3 µ ( ∇ · v ) I + f v (2) ..dimensionless formulation in primitive variables, non-conservative form where p = π − µ b ∇ · v is mean or mechanical pressure, while µ b = λ + 2 3 µ is the v = ( v 1 , v 2 ) T velocity bulk viscosity . Above equation has the same structure as (1a) while setting ( · ) T , ∇ , ∇· matrix vector transposition, gradient, divergence λ = − 2 � ∇ v + ( ∇ v ) T � 3 µ (or equivalently µ b = 0, c.f. Stokes hypothesis), but we avoid 1 D , I , unit tensor 2 t specification of µ b whose precise experimental determination is still open. Our time T ∗− T ∗ T ∗ T temperature ( T = ∞ or T = ∞ ), T ∗ W temperature on wall variable p is not thermodynamic pressure. T ∗ T ∗ W − T ∗ ∞ ρ = ρ ( T ), ρ � = ρ ( π ) density µ = µ ( T ) dynamic viscosity κ = κ ( T ) thermal conductivity c p = 1 spec. heat capacity at const. pressure, calorically perfect fluid volumetric momentum source, e.g. buoyancy f v heat source, e.g. viscous heating f T Re = L ∗ | v ∗ ∞ | ρ ∗ Reynolds number ( L ∗ is characteristic length, e.g. cyl. diameter) ∞ µ ∗ ∞ cp ∗ ∞ µ ∗ Pr = ∞ Prandtl number κ ∗ ∞ Jan Pech (Czech Acad Sci, Inst Thermomech) Temperature dependent flow 11.06.2019 4 / 18
Motivation & model Scheme Results Coupled evolutionary Incompressible Navier-Stokes-Fourier system � ∂ v � � � = −∇ p + 1 2 µ D − 2 ρ ∂ t + v · ∇ v 3 µ ( ∇ · v ) I + f v (1a) Re ∇ · ∂ρ ∂ t + ∇ · ( ρ v ) = 0 (1b) � ∂ T � 1 ρ c p ∂ t + v · ∇ T = RePr ∇ · κ ∇ T + f T (1c) ..dimensionless formulation in primitive variables, non-conservative form v = ( v 1 , v 2 ) T velocity ( · ) T , ∇ , ∇· matrix vector transposition, gradient, divergence � ∇ v + ( ∇ v ) T � 1 D , I , unit tensor 2 t time T ∗− T ∗ T ∗ T temperature ( T = ∞ or T = ∞ ), T ∗ W temperature on wall T ∗ T ∗ W − T ∗ ∞ ρ = ρ ( T ), ρ � = ρ ( π ) density µ = µ ( T ) dynamic viscosity κ = κ ( T ) thermal conductivity c p = 1 spec. heat capacity at const. pressure, calorically perfect fluid volumetric momentum source, e.g. buoyancy f v heat source, e.g. viscous heating f T Re = L ∗ | v ∗ ∞ | ρ ∗ Reynolds number ( L ∗ is characteristic length, e.g. cyl. diameter) ∞ µ ∗ ∞ cp ∗ ∞ µ ∗ Pr = ∞ Prandtl number κ ∗ ∞ Jan Pech (Czech Acad Sci, Inst Thermomech) Temperature dependent flow 11.06.2019 4 / 18
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