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Fluid dynamics of a stationary weld pool

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Abstract

Flow driven by a combination of thermocapillary, Lorentz, and buoyant forces has been investigated in an axisymmetric and stationary weld pool numerically. By assuming a small value for the capillary number, the top and bottom boundaries can be taken to be flat, and the surface deflections can be calculateda posteriori as a domain perturbation. Owing to thin boundary layers that exist at the top free surface and next to the vertical wall, very fine grids are required in these regions in order to obtain an accurate solution to the Boussinesq form of the Navier-Stokes equations. This was done by solving the governing equations by multigrid methods to which a local grid refinement technique was added. Welding of both aluminum and steel were considered. The essential difference between these two materials for this analysis is that the Prandtl number of aluminum is an order of magnitude smaller than that of steel. Through a parametric study, the thermocapillary forces and Lorentz forces were found to dominate buoyancy forces in a typical welding situation. Although the flows in weld pools include a pronounced recirculating region near the top surface, isotherms could be determined in the case of aluminum to a good approximation by a conduction analysis, owing to the smallness of its Prandtl number and the relative thinness of the welded plate considered. For steel, the isotherms deflect considerably for high current inputs.

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Abbreviations

r :

dimensionless radial coordinate

z :

dimensionless axial coordinate

L :

length of the slab

h :

thickness of the slab

u :

dimensionless radial velocity

w :

dimensionless axial velocity

p :

dimensionless pressure

T :

dimensionless temperature

f r :

radial component of Lorentz force

f z :

axial component of Lorentz force

J :

current density

B :

magnetic induction

E :

electric field strength

j r :

radial component of current density

j z :

axial component of current density

Bϕ :

azimuthal component of magnetic induction

Ψ :

stream function

ψ e :

magnetic stream function

ω :

vorticity

g :

gravitational acceleration

λ :

coefficient of volumetric expansion

v :

kinematic viscosity

μ :

dynamic viscosity

α :

thermal diffusivity

ρ :

density

k :

relative conductivity

k l :

thermal conductivity of liquid metal

k s :

thermal conductivity of solid metal

γ :

temperature coefficient of surface tension

T l :

liquidus temperature of metal

T s :

solidus temperature of metal

T ∞ :

dimensionless temperature of ambient air

μ 0 :

magnetic permeability of vacuum

μ e :

magnetic permeability of metal

κ :

electrical conductivity

ΔT :

characteristic temperature difference

U :

characteristic velocity

q 0 :

magnitude of arc heat flux

j 0 :

magnitude of current density

β :

coefficient of Gaussian heat and current flux

H :

aspect ratio

Gr:

Grashof number

Pr:

Prandtl number

Re:

Reynolds number

Re H :

magnetic pressure number

Re m :

magnetic Reynolds number

K :

conductivity ratio

X m :

melt ratio

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Authors and Affiliations

  1. Department of Chemical Engineering, Stanford University, 94305, Stanford, CA

    Natarajan Ramanan (Postdoctoral Fellow)

  2. Department of Mechanical Engineering, The Ohio State University, 43210, Columbus, OH

    Seppo A. Korpela (Associate Professor)

Authors
  1. Natarajan Ramanan
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  2. Seppo A. Korpela
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Formerly with the Department of Mechanical Engineering, The Ohio State University.

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Ramanan, N., Korpela, S.A. Fluid dynamics of a stationary weld pool. Metall Trans A 21, 45–57 (1990). https://doi.org/10.1007/BF02656423

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