Thermal convection

Whether cooling a power station, insulating a confined plasma, or simply cooling our laptop, heat transfer is one of the most important and exciting research areas in Engineering. However, several problems are still poorly understood.

We enjoy tackling old and not-so-old heat transfer problems and solve them in a creative way.

 

The thin cylinder in cross-flow

Heat transfer between a cylinder with large length to diameter ratio and a fluid stream in cross-flow occurs in many applications, such as hot-wire anemometers, submarine pipelines, electrical and telecommunication cables.

We have found that in the limit of small Péclet numbers the wire can be treated as a mathematical singularity in the plane normal to its axis:

 

 

With these approximations, the dimensionless energy equation is:

          (1)

 

Solving the energy equation with the theory of analytical functions, we have been able to obtain a solution in closed form, with no empirical coefficients:

          (2)

 

 

where K₀(x) is a modified Bessel function of second kind.

Our solution is in very good agreement with other approximate solutions existing in the literature:

 

 

V. Bertola, E. Cafaro, Analytical function theory approach to the heat transfer problem of a cylinder in cross-flow at small Péclet numbers, International Journal of Heat and Mass Transfer, Vol 49(17-18), pp. 2859-2863, 2006.

 

 

Rayleigh-Bénard convection of a viscoelastic fluid in porous media

We have studied the Horton-Rogers-Lapwood problem of Rayleigh-Bénard convection in a porous medium for a viscoelastic fluid with a single relaxation time.

 

If  is the retardation time due to the action of the porous matrix, and  is the relaxation time due to viscoelasticity, the conservation equations for mass, momentum and energy are:

 

 

with the boundary conditions  and .

 

Similar to the Salzmann-Lorenz analysis of atmospheric convection, the conservation equations can be transformed into a reduced-order dynamical system by a Galerkin projection in the space of eigenfunctions of the temperature fluctuations and the stream function (D₁ and D₂ are the dimensionless characteristic times, and r is the Rayleigh number)

 

 

 

Depending on the characteristic times, the critical Rayleigh number for the onset of convection can be significantly lower than for a Newtonian fluid:

The trajectory of the system in the phase space may converge to an equilibrium point (steady-state convection):

or to a stable orbit (oscillatory convection):

 

V. Bertola, E. Cafaro, Thermal instability of viscoelastic fluids in horizontal porous layers as initial value problem, International Journal of Heat and Mass Transfer, Vol 49(21-22), pp 4003-4012, 2006.