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Finite Element Methods for the Simulation of Incompressible Flows Weierstrass Institute for Applied Analysis and Stochastics Finite Element Methods for the Simulation of Incompressible Flows Volker John Mohrenstrasse 39 · 10117 Berlin · Germany · Tel. +49 30 20372 0 · www.wias-berlin.de · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 Outline of the Lectures 1 The Navier–Stokes Equations as Model for Incompressible Flows 2 Function Spaces For Linear Saddle Point Problems 3 The Stokes Equations 4 The Oseen Equations 5 The Stationary Navier–Stokes Equations 6 The Time-Dependent Navier–Stokes Equations – Laminar Flows Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 2 (151) 1 A Model for Incompressible Flows • conservation laws ◦ conservation of linear momentum ◦ conservation of mass • flow variables ◦ r(t;x) : density [kg=m3] ◦ v(t;x) : velocity [m=s] ◦ P(t;x) : pressure [N=m2] assumed to be sufficiently smooth in • W ⊂ R3 • [0;T] Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 3 (151) 1 Conservation of Mass • change of fluid in arbitrary volume V ¶ Z Z Z − r dx = rv · n ds = ∇ · (rv) dx ¶t V ¶V V | {z } | {z } mass transport through bdry • V arbitrary =) continuity equation rt + ∇ · (rv) = 0 • incompressibility( r = const) ∇ · v = 0 Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 4 (151) • acceleration: using first order Taylor series expansion in time (board) dv (t;x) = ¶ v(t;x) + (v(t;x) · ∇)v(t;x) dt t movement of a particle 1 Newton’s Second Law of Motion • Newton’s second law of motion net force = mass × acceleration Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 5 (151) 1 Newton’s Second Law of Motion • Newton’s second law of motion net force = mass × acceleration • acceleration: using first order Taylor series expansion in time (board) dv (t;x) = ¶ v(t;x) + (v(t;x) · ∇)v(t;x) dt t movement of a particle Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 5 (151) • principle of Cauchy: internal contact forces depend (geometrically) only on the orientation of the surface t = t(n) n – unit normal vector of the surface pointing outwards of V 1 Newton’s Second Law of Motion • acting forces on an arbitrary volume V : sum of external (body) forces ◦ gravity and internal (molecular) forces ◦ pressure ◦ viscous drag that a ’fluid element’ exerts on the ’adjacent element’ ◦ contact forces: act only on surface of ’fluid element’ Z Z F(t;x) dx + t(t;s) ds V ¶V t [N=m2] – Cauchy stress vector Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 6 (151) 1 Newton’s Second Law of Motion • acting forces on an arbitrary volume V : sum of external (body) forces ◦ gravity and internal (molecular) forces ◦ pressure ◦ viscous drag that a ’fluid element’ exerts on the ’adjacent element’ ◦ contact forces: act only on surface of ’fluid element’ Z Z F(t;x) dx + t(t;s) ds V ¶V t [N=m2] – Cauchy stress vector • principle of Cauchy: internal contact forces depend (geometrically) only on the orientation of the surface t = t(n) n – unit normal vector of the surface pointing outwards of V Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 6 (151) 1 Newton’s Second Law of Motion • it can be shown: conservation of linear momentum results in linear dependency on n t = Sn S(t;x)[N=m2] – stress tensor, dimension 3 × 3 • divergence theorem Z Z t(t;s) ds = ∇ · S(t;x) dx ¶V V • momentum equation r (vt + (v · ∇)v) = ∇ · S + F 8 t 2 (0;T]; x 2 W Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 7 (151) 1 Newton’s Second Law of Motion • model for the stress tensor ◦ torque Z Z M0 = r × F dx + r × (Sn) ds [N m] V ¶V T at equilibrium is zero =) symmetry S = S ◦ decomposition S = V + PI V [N=m2] – viscous stress tensor ◦ pressure P acts only normal to the surface, directed into V Z Z Z − Pn ds = − ∇P dx = − ∇ · (PI) dx ¶V V V Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 8 (151) 1 Newton’s Second Law of Motion • model for the stress tensor (cont.) ◦ viscous stress tensor − friction between fluid particles can only occur if the particles move with different velocities − viscous stress tensor depends on gradient of velocity − because of symmetry: on symmetric part of the gradient: velocity deformation tensor ∇v + (∇v)T (v) = D 2 − velocity not too large: dependency is linear (Newtonian fluids) 2m = 2m (v) + z − (∇ · v) V D 3 I m [kg=(m s)] – dynamic or shear viscosity z [kg=(m s)] – second order viscosity Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 9 (151) • incompressible flows: incompressible Navier–Stokes equations P F ¶t v − 2n∇ · D(v) + (v · ∇)v + ∇ = in (0;T] × W; r0 r0 ∇ · v = 0 in (0;T] × W 1 Navier–Stokes Equations • general Navier–Stokes equations r (¶t v + (v · ∇)v) 2m −2∇ · (mD(v)) − ∇ · z − 3 ∇ · vI + ∇P = F in (0;T] × W; rt + ∇ · (rv) = 0 in (0;T] × W Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 10 (151) 1 Navier–Stokes Equations • general Navier–Stokes equations r (¶t v + (v · ∇)v) 2m −2∇ · (mD(v)) − ∇ · z − 3 ∇ · vI + ∇P = F in (0;T] × W; rt + ∇ · (rv) = 0 in (0;T] × W • incompressible flows: incompressible Navier–Stokes equations P F ¶t v − 2n∇ · D(v) + (v · ∇)v + ∇ = in (0;T] × W; r0 r0 ∇ · v = 0 in (0;T] × W Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 10 (151) 1 Navier–Stokes Equations • Claude Louis Marie Henri Navier (1785 – 1836) George Gabriel Stokes (1819 – 1903) Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 11 (151) 1 Dimensionless Incompressible Navier–Stokes Equations • dimensionless equations needed for (numerical) analysis and numerical simulations • reference quantities of flow problem ◦ L [m] – a characteristic length scale ◦ U [m=s] – a characteristic velocity scale ◦ T ∗ [s] – a characteristic time scale • transform of variables x0 v t0 x = ; u = ; t = L U T ∗ • rescaling L 2n P L ∗ ¶t u − ∇ · D(u) + (u · ∇)u + ∇ 2 = 2 F in (0;T] × W; UT UL r0U r0U ∇ · u = 0 in (0;T] × W; Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 12 (151) 1 Dimensionless Incompressible Navier–Stokes Equations • defining P UL L L p = 2 ; Re = ; St = ∗ ; f = 2 F r0U n UT r0U p – new pressure Re – Reynolds number St – Strouhal number f – new right hand side • result 2 St¶ u − ∇ · (u) + (u · ∇)u + ∇p = f in (0;T] × W; t Re D ∇ · u = 0 in (0;T] × W • generally T ∗ = L=U =) St = 1 Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 13 (151) 1 Dimensionless Incompressible Navier–Stokes Equations • dimensionless Navier–Stokes equations ◦ conservation of linear momentum ◦ conservation of mass −1 T ut − 2Re ∇ · D(u) + ∇ · (uu ) + ∇p = f in (0;T] × W ∇ · u = 0 in [0;T] × W u(0;x) = u0 in W + boundary conditions • given: • to compute: d ◦ W ⊂ R ;d 2 f2;3g: domain ◦ velocity u, with ◦ T : final time ∇u + ∇uT ◦ u : initial velocity (u) = ; 0 D 2 ◦ boundary conditions velocity deformation tensor ◦ pressure p • parameter: Reynolds number Re Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 14 (151) 1 The Reynolds Number • Reynolds number LU Re = n convective forces = viscous forces Osborne Reynolds (1842 – 1912) • rough classification of flows: ◦ Re small: steady-state flow field (if data do not depend on time) ◦ Re larger: laminar time-dependent flow field ◦ Re very large: turbulent flows Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 15 (151) • alternative expression of viscous term (due to ∇ · u = 0) 2∇ · D(u) = Du • alternative expression of convective term (due to ∇ · u = 0) (u · ∇)u = ∇ · (uuT ) 1 Dimensionless Incompressible Navier–Stokes Equations • simplified form (for mathematics) ¶t u − 2n∇ · D(u) + (u · ∇)u + ∇p = f in (0;T] × W; ∇ · u = 0 in (0;T] × W n = Re−1 – dimensionless viscosity Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 16 (151) 1 Dimensionless Incompressible Navier–Stokes Equations • simplified form (for mathematics) ¶t u − 2n∇ · D(u) + (u · ∇)u + ∇p = f in (0;T] × W; ∇ · u = 0 in (0;T] × W n = Re−1 – dimensionless viscosity • alternative expression of viscous term (due to ∇ · u = 0) 2∇ · D(u) = Du • alternative expression of convective term (due to ∇ · u = 0) (u · ∇)u = ∇ · (uuT ) Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02.
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