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Fractional four-step finite element method for analysis of thermally coupled fluid-solid interaction problems

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Abstract

An integrated fluid-thermal-structural analysis approach is presented. In this approach, the heat conduction in a solid is coupled with the heat convection in the viscous flow of the fluid resulting in the thermal stress in the solid. The fractional four-step finite element method and the streamline upwind Petrov-Galerkin (SUPG) method are used to analyze the viscous thermal flow in the fluid. Analyses of the heat transfer and the thermal stress in the solid are performed by the Galerkin method. The second-order semiimplicit Crank-Nicolson scheme is used for the time integration. The resulting nonlinear equations are linearized to improve the computational efficiency. The integrated analysis method uses a three-node triangular element with equal-order interpolation functions for the fluid velocity components, the pressure, the temperature, and the solid displacements to simplify the overall finite element formulation. The main advantage of the present method is to consistently couple the heat transfer along the fluid-solid interface. Results of several tested problems show effectiveness of the present finite element method, which provides insight into the integrated fluid-thermal-structural interaction phenomena.

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Abbreviations

g :

gravitational acceleration

Gr :

Grashofnumber, Gr = gβ T (T h − T c)L 3/ν 2 (ratio of buoyancy to viscous force)

k :

thermal conductivity

Nu :

local Nusselt number (ratio of convective to conductive heat transfer across (normal to) boundary)

\(\overline {Nu}\) :

average Nusselt number

Pe :

Pecletnumber

p :

fluid pressure

Pr :

Prandtl number, Pr = ν/α (ratio of momentum diffusivity (kinematic viscosity) to thermal diffusivity)

\(\tilde Q\) :

heat generation per unit volume

Re :

Reynolds number, Re = ρυd/µ (ratio of inertial force to viscous force)

t :

time

T :

temperature

u 1 :

velocity component in the x-direction

u 2 :

velocity component in the y-direction

υ 1 :

solid displacement in the x-direction

υ 2 :

solid displacement in the y-direction

x :

horizontal distance

y :

vertical distance

α :

thermal diffusivity, α = k/ρc

α sf :

solid-to-fluid thermal diffusivity ratio, α sf = α s/α f

β f :

thermal expansion coefficient

K sf :

solid-to-fluid thermal conductivity ratio, K sf = k s/k f

ν :

kinematic viscosity, ν = μ/ρ

ρ :

density

i, j :

nodal quantities

int:

interface

o:

cold surfaces

f:

fluid

s:

solid

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Authors and Affiliations

  1. Computer-Aided Research Unit, National Metal and Materials Technology Center, Pathumthani, 12120, Thailand

    A. Malatip

  2. Department of Mechanical Engineering, Faculty of Engineering, Chulalongkorn University, Patumwan, Bangkok, 10330, Thailand

    N. Wansophark & P. Dechaumphai

Authors
  1. A. Malatip
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  2. N. Wansophark
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  3. P. Dechaumphai
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Corresponding author

Correspondence to P. Dechaumphai.

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Malatip, A., Wansophark, N. & Dechaumphai, P. Fractional four-step finite element method for analysis of thermally coupled fluid-solid interaction problems. Appl. Math. Mech.-Engl. Ed. 33, 99–116 (2012). https://doi.org/10.1007/s10483-012-1536-9

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  • DOI: https://doi.org/10.1007/s10483-012-1536-9

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