Numerical Simulation of Newtonian and Non-Newtonian Flows in Bypass Vladimír Prokop, Karel Kozel Czech Technical University Faculty of Mechanical Engineering Department of Technical Mathematics Vladimír Prokop, Karel Kozel (CTU) Bypass Flows / 33
Outline Motivation Non-Newtonian behavior Blood flow Power-law fluids 2 Mathematical model Governing equations for incompressible laminar flows in 2D Governing equations for incompressible laminar flows in 3D Boundary conditions 3 Numerical model Numerical solution by Finite volume method 4 Results 5 Conclusion Vladimír Prokop, Karel Kozel (CTU) Bypass Flows 2 / 33
Motivation Non-Newtonian Fluids Many common fluids are non-newtonian: paints solutions of various polymers food products Applications: biomedicine food industry chemistry glaciology Vladimír Prokop, Karel Kozel (CTU) Bypass Flows 3 / 33
Non-Newtonian behavior Main points of non-newtonian behavior: the ability of the fluid to shear thin or shear thicken in shear flows the presence of non-zero normal stress differences in shear flows the ability of the fluid to yield stress the ability of the fluid to exhibit relaxation the ability of the fluid to creep Figure: Non-Newtonian fluid behavior Vladimír Prokop, Karel Kozel (CTU) Bypass Flows 4 / 33
Blood flow Blood is considered as a continuum Blood is non-newtonian suspension of cells in plasma Experimental tests reveal that blood exhibits non-newtonian phenomena such as shear thinning, creep and stress relaxation Only shear thinning is considered in this work It is reasonable to model it as a Newtonian fluid in greater vessels Can be described by conservation laws of mass and momentum The pulsatile character of blood flow is not considered as well as the elasticity of arterial walls Vladimír Prokop, Karel Kozel (CTU) Bypass Flows 5 / 33
Arterial flow phenomena Arterial flow phenomena in atherosclerotic vessel separation recirculation secondary flow motion Figure: Atherosclerotic vessel Figure: Atherosclerotic vessel Vladimír Prokop, Karel Kozel (CTU) Bypass Flows 6 / 33
Power-law fluids Power-law fluids τ(e) = 2ν 0 e r e, τ is the stress tensor e = (e ij ), i,j =,2 is the strain tensor with components e = u x, e 2 = e 2 = (v x + u y )/2, e 22 = v y e denotes the Euclidean norm of a tensor ν 0 is a positive constant related to the limit of generalized viscosity µ g r is a constant of the model Vladimír Prokop, Karel Kozel (CTU) Bypass Flows 7 / 33
Navier-Stokes equations The generalized system of 2D Navier-Stokes equations and continuity equation for incompressible laminar flows in conservative dimensionless form Inviscid fluxes RW t + F i x + Gi y = R Re (F v x + Gv y ), u v F i = u + p,g i = uv uv v 2 + p R = diag 0,, Reynolds number in 2D defined as Re = dq /ν Quantity q is a characteristic velocity (the speed of upstream flow) ν = η/ρ is the kinematic viscosity d is a length scale (the width of the channel) W = (p,u,v) T is the vector of solution Vladimír Prokop, Karel Kozel (CTU) Bypass Flows 8 / 33
Navier-Stokes equations The dimensionless quantities components of the velocity vector u = u /q,v = v /q pressure p = p /ρq 2 Newtonian viscous fluxes 0 0 F v = u x, G v = u y v x v y Non-Newtonian viscous fluxes 0 0 F v = 2 e r u x, G v = 2 e r (u y + v x ) 2 e r (v x + u y ) 2 e r v y Vladimír Prokop, Karel Kozel (CTU) Bypass Flows 9 / 33
Navier-Stokes equations The generalized system of 3D Navier-Stokes equations and continuity equation for incompressible laminar flows in conservative dimensionless form RW t + Fx i + Gy i + Hz i = R Re (F x v + Gy v + Hz v ), R = diag 0,, Inviscid and viscous fluxes u v w F i = u 2 + p uv,gi = uv v 2 + p Hi = uw vw uw vw w 2 + p Reynolds number in 3D defined as Re = d h q /ν Quantity q is a characteristic velocity (the speed of upstream flow) ν = η/ρ is the kinematic viscosity d h = 4S/O is the hydraulic diameter S is the area section of the duct O is the wetted perimeter Vladimír Prokop, Karel Kozel (CTU) T Bypass Flows 0 / 33
Navier-Stokes equations The dimensionless quantities components of the velocity vector u = u /q,v = v /q,w = w /q pressure p = p /ρq 2 Newtonian viscous fluxes 0 0 0 F v = u x v x, Gv = u y v y, Hv = u z v z w x w y w z Vladimír Prokop, Karel Kozel (CTU) Bypass Flows / 33
Boundary conditions At the inlet the Dirichlet boundary condition for the velocity components (u,v) = (q,0) is prescribed and the pressure p is computed by extrapolation from a domain. At the outlet the value of the pressure is prescribed by p = p 2, where p 2 is the dimensionless value of the pressure, that is lower then the initial value of the pressure at the inlet to ensure pressure gradient. The velocity components are extrapolated at the outlet. On the walls one considers the non-permeability and no-slip conditions for the velocity and the value of the pressure is taken from inside of the domain. u=v=0 q p=p 2 Vladimír Prokop, Karel Kozel (CTU) Bypass Flows 2 / 33
Finite volume method A steady state solution is considered Use of the artificial compressibility method The continuity equation is completed with the term p t /a 2, where a 2 > 0. The pressure satisfies the artificial equation of state: p = ρ/δ, in which ρ is the artificial density, δ is the artificial compressibility, that is connected to the artificial speed of sound by relation a = δ 2 Governing equations has the form W t + Fx i + Gi y = R ( F v Re x + Gy v ), where W = (p/a 2,u,v) T After rewriting the previous equation W t = ( F x + G y ), where F = F i Re F v, G = G i Re Gv Vladimír Prokop, Karel Kozel (CTU) Bypass Flows 3 / 33
Finite volume method 2D The system of equations is integrated over a finite volume D ij W t dxdy = ( Fx + G ) y dxdy. D ij D ij Mean value and Green s theorem are applied W t ij = Fdy Gdx, µij = µ ij D ij dxdy. Velocity derivatives are computed using dual volume cells The integral on the right hand side is numerically approximated by D ij W t ij = µ ij 4 F ij,k y k G ij,k x k. k= Vladimír Prokop, Karel Kozel (CTU) Bypass Flows 4 / 33
Finite volume method 3D Governing equations has the form W t +Fx+G i y+h i z i = R ( F v Re x + Gy v + Hz v ), where W = (p/a 2,u,v,w) T After rewriting the previous equation W t = ( F x + G y + H z ), where F = F i Re F v, G = G i Re Gv, H = G i Re Hv The system of equations is integrated over a finite volume D ij W t dxdy = ( Fx + G y + H z ) dv. D ijk D ij Vladimír Prokop, Karel Kozel (CTU) Bypass Flows 5 / 33
Finite volume method 3D Mean value and Green s theorem are applied W t ijk = ( F, µ G, H) ln 0 l S l = ( F, ijk µ G, H) l n l, ijk D ijk D ijk µ ijk = dv. D ijk n l = nl 0 S l, where nl 0 is unit normal vector and S l is a volume of a side of finite volume cell Velocity derivatives are computed using dual volume cells The integral on the right hand side is numerically approximated by W t ijk = µ ijk 6 ( Fijk,l n l + G ijk,l n 2l + H ) ijk,l n 3l l= Vladimír Prokop, Karel Kozel (CTU) Bypass Flows 6 / 33
Finite volume method geometry of basic finite volume cell (2D) numerical approximation of spatial part D i,j+ W t ij = µ ij 4 F k y k G k x k k= m4 D i,j A4 k 4 m m 2 k 3 m 3 D ij 00 00 00 00 0 A k 2 A D ij A 3 k 2 D i+,j numerical approximation of derivatives in viscous fluxes u x = µ d d u y = µ d udy 4 u m y m Figure: Basic finite volume cell d m=... Vladimír Prokop, Karel Kozel (CTU) Bypass Flows 7 / 33 m= udx 4 u m x m
Finite volume method Three-stage Runge-Kutta method W n i,j = W (0) i,j W (r) i,j = W (0) i,j α r trw (r ) i,j, r =,...,m W n+ i,j RW (r ) i,j = W (m) i,j,m = 3, = RW (r ) i,j DW n i,j α = 0.5,α 2 = 0.5,α 3 =.0 DW n ij is the artificial viscosity term of Jameson s type Form of residual (2D and 3D) RW ij = µ ij RW ijk = 4 k= ( Fij,k y k G ij,k x k ), 6 ) ( F ijk,l n l + G ijk,l n 2l + H ijk,l n 3l µ ijk Vladimír Prokop, Karel Kozel (CTU) l= Bypass Flows 8 / 33
Artificial viscosity artificial viscosity term (Jameson s type, 2D) D (2) W ij = Eγ i (W i+,j 2W i,j + W i,j ) + Eγ j (W i,j+ 2W i,j + W i,j ) E = diag ǫ,ǫ 2,ǫ 3,ǫ ǫ 2,ǫ 3 R,γ i = max(γ i,γ i2 ),γ j = max(γ j,γ j2 ) γ i = p i+,j 2p i,j + p i,j p i+,j + 2p i,j + p i,j, γ i2 = p i,j 2p i,j + p i 2,j p i,j + 2p i,j + p i 2,j, γ j = p i,j+ 2p i,j + p i,j p i,j+ + 2p i,j + p i,j, γ j2 = p i,j 2p i,j + p i,j 2 p i,j + 2p i,j + p i,j 2. time step t = min i,j,k CFLµ i,j ρ A y k + ρ B x k + 2 Re ( x 2 ), k + yk 2 µ i,j Vladimír Prokop, Karel Kozel (CTU) Bypass Flows 9 / 33
Comparison of isolines of velocity, Re = 500 2 Re=500, Non-Newtonian.5 Q: 0. 0.3 0.5 0.7 0.8 0.9..2.3.4.5.6.7 0.5 0 0.3 0..4.2.. 0.9.3 0.5 0..4.5.7.2.3 0.7 0.9 0. 0.8 0.8 0.5 0.7 0.3.6 0.7.4.5-0.5 0.7 2 4 6 8 X 0.3 0. 0.5 2.5 Re=500, Newtonian Q: 0. 0.3 0.5 0.7 0.8 0.9..2.3.4.5.6.7 0. 0. 0.5 0-0.5 0.9. 0.3 0.8.2 0.7.3.3 0.5 0.5 0.3 0.5 0.3 0.5.5 0.7.2 0. 2 4 6 8 0 X.4 0.3. 0.3 0.9 0.7.4 0.9 0.3 Vladimír Prokop, Karel Kozel (CTU) Bypass Flows 20 / 33
Comparison of vector field of velocity, Re = 500 2 Re=500, Non-Newtonian.5 Q: 0. 0.3 0.5 0.7 0.8 0.9..2.3.4.5.6.7 0.5 0-0.5 2 4 6 8 X 2.5 Re=500, Newtonian Q: 0. 0.3 0.5 0.7 0.8 0.9..2.3.4.5.6.7 0.5 0-0.5 2 4 6 8 0 X Vladimír Prokop, Karel Kozel (CTU) Bypass Flows 2 / 33
Velocity magnitude profiles, Re=500.6 velocity magnitude, x=5 bypass, channel Re=500 non-newtonian Newtonian.4.2 Q 0.8 - - 0 Y Vladimír Prokop, Karel Kozel (CTU) Bypass Flows 22 / 33
Convergence, Re=500-2 -4 rez u, channel, Re=500, non-newtonian -2-4 rez u, bypass, Re-500, non-newtonian -6-6 rez u -8 rez u -8-0 -0-2 -2-4 0 50000 00000 50000 200000 250000 steps 0 50000 00000 50000 200000 250000 steps -2-4 rez u, channel, Re=500, Newtonian -2-4 rez u, bypass, Re=500, Newtonian -6-6 rez u -8 rez u -8-0 -0-2 -2-4 0 0000 20000 30000 40000 steps 0 0000 20000 30000 40000 steps Vladimír Prokop, Karel Kozel (CTU) Bypass Flows 23 / 33
Comparison of isolines of velocity, Re = 200 Re=200, Non-Newtonian Q: 0.8.2.4.6.8 2 2.2 2.4 2.6 0.8.4.2.4.2 0.8.6.2.2 0.8.4 0.2 0.8.2.2 0.8.2 0.8.4 0.8.4 2 4 6 8 0 X Re=200, Newtonian Q: 0.8.2.4.6.8 2 2.2 2.4 2.6.4.4 0.8.6.2.4 0.8.2.2.8 0.8.2 2 0.4 0.8.4.4 0.8.2.4 0.8 2 4.2 6 8 X Vladimı r Prokop, Karel Kozel (CTU) Bypass Flows 24 / 33
Comparison of vector field of velocity, Re = 200 Re=200, Non-Newtonian Q: 0.8.2.4.6.8 2 2.2 2.4 2.6 0 2 4 6 8 0 X Re=200, Newtonian Q: 0.8.2.4.6.8 2 2.2 2.4 2.6 0 2 4 6 8 X Vladimír Prokop, Karel Kozel (CTU) Bypass Flows 25 / 33
Velocity magnitude profiles, Re=200 2.5 2 velocity magnitude, x=5 bypass, channel Re=200 non_newtonian Newtonian.5 Q 0.5 0-0.8 - - - 0 Y Vladimír Prokop, Karel Kozel (CTU) Bypass Flows 26 / 33
Convergence, Re=200 0 - rez u, channel, Re=200, non-newtonian -2 rez u, bypass, Re=200, non-newtonian -2 rez u rez u -3-3 -4-4 -5 0 200000 400000 600000 steps 0 200000 400000 600000 steps - -2-2 rez u, channel, Re=200, Newtonian -3 rez u, bypass, Re=200, Newtonian -3 rez u -4-5 rez u -4-5 -6-6 -7-7 0 00000 200000 300000 400000 steps 0 00000 200000 300000 400000 steps Vladimír Prokop, Karel Kozel (CTU) Bypass Flows 27 / 33
3D angular bypass, Re=500 Q.8.6.4.2. 0.8 0. Re=500, main channel Re=500 Vladimı r Prokop, Karel Kozel (CTU) Bypass Flows Q 6 0.36 0.3 6 0. 0.06 0. 28 / 33
Unsteady flow in bypass, Re=000 The outlet pressure is prescribed by sinus function in form: p 2 = p 20 ( + α sin2πωt), where ω is a frequency and α is an amplitude Re=000, t=00,0,s=4 Re = 000,t=.6,p=4 Vladimír Prokop, Karel Kozel (CTU) Bypass Flows 29 / 33
Unsteady flow in bypass, Re=000 Re=000,t=03.2,p=4 Re=000,t=04.8,p=4 Re=000,t=06.4,p=4 Vladimír Prokop, Karel Kozel (CTU) Bypass Flows 30 / 33
Unsteady flow in bypass, Re=500 Re=500, s=40,t=0 per=4 Re=500, s=40,t=00.8 per=4 Re=500, s=40, t=.4 per=4 Vladimír Prokop, Karel Kozel (CTU) Bypass Flows 3 / 33
Unsteady flow in bypass, Re=500 Re=500, s=40, t=02.0 per=4 Re=500, s=40, t=02.6 per=4 Re=500,s=40, t=03.2 per=4 Vladimír Prokop, Karel Kozel (CTU) Bypass Flows 32 / 33
Conclusion Numerical model of Newtonian and non-newtonian flow was implemented The model was tested on diffrent geometries and for different Reynolds numbers Next steps Use of different non-newtonian models Increasing mesh quality Extension of the model to 3D Unsteady computation Vladimír Prokop, Karel Kozel (CTU) Bypass Flows 33 / 33