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2094 Finite Element Analysis of Solids and Fluids

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					MIT OpenCourseWare http://ocw.mit.edu

Spring 2008

2.094 Finite Element Analysis of Solids and Fluids

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2.094 — Finite Element Analysis of Solids and Fluids

Fall ‘08

Lecture 17 - Incompressible fluid flow and heat transfer, cont’d
Prof. K.J. Bathe MIT OpenCourseWare

17.1

Abstract body
Reading: Sec. 7.4

Fluid Flow Sv , Sf Sv ∪ Sf = S Sv ∩ Sf = 0

Heat transfer Sθ , Sq Sθ ∪ Sq = S Sθ ∩ Sq = 0

17.2

Actual 2D problem (channel flow)

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MIT 2.094

17. Incompressible fluid flow and heat transfer, cont’d

17.3

Basic equations

P.V. velocities � � � � v i ρvi,j vj dV + τij eij dV = v i fiB dV +
V V V Sf

v i f fi f dSf

S

S

(17.1)

Continuity � pvi,i dV = 0
V

(17.2)

P.V. temperature � � � � θρcp θ,i vi dV + θ,i kθ,i dV = θq B dV +
V V V Sq

θ q S dS

S

(17.3)

F.E. solution xi = vi = θ= p= � � � � hk xk i
k hk vi

(17.4) (17.5) (17.6) (17.7)

hk θk ˜ hk pk

⇒ F (u) = R


⎞ v
 u = ⎝ p ⎠ nodal variables θ

⎛

(17.8)

17.4

Model problem
dθ d2 θ =k 2 dx dx

1D equation, ρcp v (17.9)

(v is given, unit cross section) Non-dimensional form (Section 7.4) Pe dθ d2 θ = 2 dx dx 72 (17.10)

MIT 2.094

17. Incompressible fluid flow and heat transfer, cont’d

Pe =

vL , α

α=

k ρcp θ∗ is non-dimensional � � exp Pe x − 1
 θ − θL L = exp (Pe) − 1 θR − θ L

(17.11)

(17.12)


(17.10) in F.E. analysis becomes � � dθ dθ dθ θPe dV + dV = 0 dx V V dx dx Discretized by linear elements:

(17.13)

h∗ =

h L

θ(ξ) =

� � ξ ξ 1 − ∗ θi−1 + ∗ θi h h

(17.14)

For node i:
 −θi−1 − where Pee = vh α � � h = Pe L (17.16) Pee Pee
 θi−1 + 2θi − θi+1 + θi+1 = 0 2 2 (17.15)

This result is the same as obtained by finite differences � 1 � θ�� � = 2 (θi+1 − 2θi + θi−1 ) i (h∗ ) � θi+1 − θi−1 � θ� � = 2h∗ i 73

(17.17) (17.18)

MIT 2.094 Considered θi+1 = 1, θi−1 = 0. Then θi = 1 − (Pee /2) 2

17. Incompressible fluid flow and heat transfer, cont’d

(17.19)

Physically unrealistic solution when Pee > 2. For this not to happen, we should refine the mesh—a very fine mesh would be required. We use “upwinding” dθ � θi − θi−1 � � = dx i h∗ The result is (−1 − Pee ) θi−1 + (2 + Pee ) θi − θi+1 = 0 Very stable, e.g. � θi−1 = 0 θi+1 = 1 (17.21) (17.20)

⇒ θi =

1 2 + Pee

(17.22)

Unfortunately it is not that accurate. To obtain better accuracy in the interpolation for θ, use the function � x� exp Pe L − 1 (17.23) exp (Pe) − 1 The result is Pee dependent:

This implies flow-condition based interpolation. We use such interpolation functions—see references.

References
[1] K.J. Bathe and H. Zhang. “A Flow-Condition-Based Interpolation Finite Element Procedure for Incompressible Fluid Flows.” Computers & Structures, 80:1267–1277, 2002. [2] H. Kohno and K.J. Bathe. “A Flow-Condition-Based Interpolation Finite Element Procedure for Triangular Grids.” International Journal for Numerical Methods in Fluids, 51:673–699, 2006.

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MIT 2.094	

17. Incompressible fluid flow and heat transfer, cont’d

17.5

FSI briefly


Lagrangian formulation for the structure/solid Arbitrary Lagrangian-Eulerian (ALE) formulation Let f be a variable of a particle (e.g. f = θ). Consider 1D � ∂f ∂f � f˙� = + v (17.24) ∂t ∂x particle where v is the particle velocity. For a mesh point, � ∂f ∂f � f ∗� = + vm	 ∂t ∂x mesh point where vm is the mesh point velocity. Hence, � � ∂f � � f˙� = f ∗� + (v − vm )	 ∂x particle mesh point

(17.25)

(17.26)

Use (17.26) in the momentum and energy equations and use force equilibrium and compatibility at the FSI boundary to set up the governing F.E. equations.

References
[1] K.J. Bathe, H. Zhang and M.H. Wang. “Finite Element Analysis of Incompressible and Compressible Fluid Flows with Free Surfaces and Structural Interactions.” Computers & Structures, 56:193–213, 1995. [2] K.J.	 Bathe, H. Zhang and S. Ji. “Finite Element Analysis of Fluid Flows Fully Coupled with Structural Interactions.” Computers & Structures, 72:1–16, 1999. [3] K.J. Bathe and H. Zhang. “Finite Element Developments for General Fluid Flows with Structural Interactions.” International Journal for Numerical Methods in Engineering, 60:213–232, 2004. 75


				
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