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Convection-Type Problems
Published in Gianni Comini, Stefano Del Giudice, Carlo Nonino, Finite Element Analysis in Heat Transfer, 2018
Gianni Comini, Stefano Del Giudice, Carlo Nonino
The vorticity equation consists only of the unsteady term ϱ ∂w/∂ϑ, the advective term ϱ(u ∂w/∂x + v ∂w/∂y) and the diffusion term ∂/∂x (μ ∂w/∂x) + ∂/∂y (μ ∂w/∂y) since the pressure has been eliminated. Equation (7.24) can be solved only if the flow field has been determined, and, consequently, we need at least one additional equation, such as the stream function equation, to compute the flow field. In fact, defining the stream function ψ by its first derivatives () ∂ψ∂y=u
Numerical Methods for Convection Heat Transfer
Published in Yogesh Jaluria, Kenneth E. Torrance, Computational Heat Transfer, 2017
Yogesh Jaluria, Kenneth E. Torrance
A very commonly used method for the vorticity equation is the ADI method, discussed in Chapter 5. This method, originally given by Peaceman and Rachford (1955), employs the inversion of tridiagonal matrices which considerably simpli es the solution. The method is implicit only in the x direction for the rst half time step and then only in the y direction for the successive halftime step, with the actual time step split into two, each of duration Δ/2. The intermediate values obtained after the rst half-step are then employed for the second half-step to yield the values at the end of the time interval Δt. The method is unconditionally stable for the diffusion problem and, being implicit in only one direction for each computation, gives rise to a tridiagonal matrix for each case, instead of a block-tridiagonal form which arises when other implicit methods are employed for two-dimensional problems. The truncation error for this method, when employed for a linear equation is 0[(Δt)2,(Δx)2,(Δy)2]. For the vorticity equation, the linearized problem is considered and the ADI method is applied in two halftime steps of Δt/2 each as:
Barotropic instability of a zonal jet on the sphere: from non-divergence through quasi-geostrophy to shallow water
Published in Geophysical & Astrophysical Fluid Dynamics, 2021
Nathan Paldor, Ofer Shamir, Chaim. I. Garfinkel
The non-divergent (ND) model reduces the order of the shallow water model from third-order in time to first-order in time by replacing the horizontal momentum equations with the vorticity-divergence ones and setting the divergence equal to zero. The resulting system consists of a single equation for the vorticity, namely the non-divergent barotropic vorticity equation, and two constraints: A boundary value problem, referred to as the ‘balance equation’, which relates the vorticity, horizontal velocity and free-surface height, and a constant (with respect to time) free-surface height. The resulting system after setting the divergence equal to zero is where is the horizontal velocity vector, is the unit vector in the vertical direction and is the relative vorticity. Note that the ND model in (11) is obtained from the SWEs model by taking the curl and divergence of the momentum equations and neglecting and compared to H, which implies that the two models can be expected to converge in the asymptotic limit (see also Cho and Polvani 1996).
On the dynamics of an idealised bottom density current overflowing in a semi-enclosed basin: mesoscale and submesoscale eddies generation
Published in Geophysical & Astrophysical Fluid Dynamics, 2020
Mathieu Morvan, Xavier Carton, Pierre L'Hégaret, Charly de Marez, Stéphanie Corréard, Stéphanie Louazel
The main source terms in the vorticity equation are the bottom pressure torque and the bottom drag curl (Gula et al.2015). Along the southern coast of the basin, the intensity of the bottom pressure torque and the bottom drag curl substantially decreases (see figure 11(a)). Their minimum values are reached for . At this location the mean flow interaction with the sloping bathymetry is weak. Indeed, it corresponds to the location of the formation of the dipole cyclone/anticyclone. To highlight the primary instability generating the dipolar eddy, we compute the energy conversion term averaged over the period of its formation, and integrated between 300 and 600 m depth (see figure 11(b)). The vertical buoyancy flux prevails on the horizontal and the vertical shear production. The vertical buoyancy flux has a positive signal where the baroclinic instability is triggered. This means that the mean potential energy of the BBC is converted into eddy kinetic energy. The BBC undergoes baroclinic instability as it flows over the southern sloping bathymetry resulting in the formation of a mesoscale dipole cyclone/anticyclone.