Symbolic analysis of reduced forms of the Navier-Stokes equations

AbstractA unified development of symbolic analysis is presented. Symbolic analysis is used to identify reduced forms of the linearized steady Navier-Stokes equations which permit computational solutions to be obtained in a single spatial march in the dominant flow direction. In particular it is demonstrated that the “parabolized” form of the Navier-Stokes equations, although not parabolic, is well-posed as an initial-value problem in space, provided the solution is restricted to functions with compact support. The effectiveness of symbolic analysis for determining the well-posedness of complex systems of equations, such as the Navier-Stokes equations, is clearly demonstrated

Scalings for unsteady natural convection boundary layer under time-varying heating flux in a small Prandtl number fluid

The unsteady natural convection boundary layer (NCBL) on a vertical wall heated by time varying flux in initially quiescent homogeneous fluid with a small Prandtl number (Pr) was studied. Scalings for the parameters typifying NCBL behavior, including plate temperature, maximum vertical velocity, thermal boundary-layer thickness, and velocity boundary-layer thickness, at different development stages, and the time for the transition from the start-up stage to the quasi-steady state, were developed by scaling analysis. The obtained scalings were compared to and validated by the numerical results with different values of Pr, the Rayleigh number Ra and the dimensionless time-varying heat flux frequency fn, over 106 ≤ Ra ≤ 109, 0.01 ≤ Pr ≤ 0.5, and 0.001 ≤ fn ≤ 0.025. It is also found that the development of the boundary layer at the start-up stage is one-dimensional and but becomes two-dimensional at the quasi-steady state

Solving the Navier-Stokes equations on a workstation cluster

A Finite Volume CFD code for modelling Natural convection has been parallelised using High Performance Fortran (HPF). A comparison is made between using HPF and message passing libraries (PVM and MPI) in terms of performance and ease of conversion. For the code being used by the authors HPF and MPI give similar speeds, and HPF was considered significantly simpler to program

Efficient Algorithm on a Non-staggered Mesh for Simulating Rayleigh-Benard Convection in a Box

An efficient semi-implicit second-order-accurate finite-difference method is described for studying incompressible Rayleigh-Benard convection in a box, with sidewalls that are periodic, thermally insulated, or thermally conducting. Operator-splitting and a projection method reduce the algorithm at each time step to the solution of four Helmholtz equations and one Poisson equation, and these are are solved by fast direct methods. The method is numerically stable even though all field values are placed on a single non-staggered mesh commensurate with the boundaries. The efficiency and accuracy of the method are characterized for several representative convection problems.Comment: REVTeX, 30 pages, 5 figure

Direct simulation of fountains with intermediate Froude and Reynolds numbers

Fountains with intermediate Froude and Reynolds numbers are of fundamental interest, especially for understanding the mechanism of turbulence and entrainment in turbulent fountains. However there has been little work done to investigate the behavior of these transitional fountains. In this work, the transient behavior of axisymmetric fountains with 1?Fr?8 and 200?Re?800 is studied by direct numerical simulation. It is found that when Re?200, there is little mixing between the downflow of the fountain and the ambient, even for Fr as high as 8; however, when Re?400, the mixing between the fountain downflow and the ambient becomes stronger, indicating that the entrainment and the turbulence are mainly due to Re and the contribution from Fr is much smaller; The maximum fountain penetration height z m fluctuates, even when the flow reaches the quasi-steady state, due to the interaction between the downflow and the bottom boundary and/or the entrainment between the downflow and the ambient; The averaged z m scales with FrRe 1/4 for 1?Fr?8 and 100?Re?800.\u

Unified Prandtl number scaling for start-up and fully developed natural-convection boundary layers for both Pr ≳1 and Pr ≳1 fluids with isothermal heating

Recent studies have used scaling analysis to obtain simple power-law relations that accurately predict the Prandtl (Pr) number dependency of natural-convection boundary layers subjected to both isothermal and ramped heating conditions, when Pr>1. The analysis used in those studies cannot be extended to Pr<1 fluids, and it is not clear at present whether such simple scaling relations can be developed for Pr<1 fluids. In the present study, the Pr>1 scalings are shown to perform well for the start-up stage of the Pr<1 flow, but not for the fully developed flow. The Pr>1 scalings are modified to provide unified Prandtl number scalings for fully developed natural-convection boundary layers for both Pr≳1 and Pr≲1, with the unknown powers obtained empirically via direct numerical simulation. The modified scalings are shown to perform well for the fully developed flow, with the exception being the prediction of the inner viscous boundary-layer thicknes

Onset of entrainment in transitional round fountains

It is of fundamental interest to understand the behavior of transitional fountains with intermediate Froude and Reynolds numbers, together with the associated entrainment and turbulence. In this work, the transient behavior of axisymmetric fountains with 1 6 Fr 6 8 and 200 6 Re 6 800 is studied by direct numerical simulation. It is found that at Re 6 200, there is little entrainment present at the upflow–downflow interface and at the downflow–ambient interface, even for a value of Fr as high as 8; however, at Re > 200, entrainment is present at these interfaces and the extent increases with Re, which clearly demonstrates that entrainment is strongly dependent on Re whereas the contribution from the Fr effect is relatively much smaller. The DNS results also show that zm, which is the maximum fountain penetration height, fluctuates, even when the flow reaches full development, due to the entrainment at the upflow–downflow and the downflow–ambient interfaces, and the averaged zm scales with Fr32Re14 for 1 < Fr 6 8 and 100 6 Re 6 800

Unsteady natural convection on an evenly heated vertical plate for Prandtl number Pr<1

The transient behavior of the natural convection boundary-layer flow adjacent to a vertical plate heated with a uniform flux in a quiescent homogeneous ambient fluid with Prandtl number Pr<1 is investigated by scaling analysis and direct numerical simulation (DNS). The flow is characterized by a startup stage, a short transitional stage and a steady state. The flow is parametrized by the thermal and velocity boundary-layer thickness scales, the vertical velocity scale, the time scale for the boundary layer to reach the steady state and the plate temperature scale. Scaling analysis is used to obtain laws relating these quantities to the flow governing parameters, the Rayleigh number Ra, the Prandtl number, and the Boussinesq number Bo=RaPr which is a much more important control parameter than Ra for small Pr fluids. A series of DNS with selected values of Ra and Pr in the ranges of 106≤Ra≤1010 and 0.01≤Pr≤0.5 are used to validate the scaling laws and obtain scaling constants

Start-up flow on a vertical semi-infinite heated plate

The start-up of natural convection boundary layer flow on a semi-infinite heated plate is simulated using a time-accurate finite-volume Navier-Stokes solver. The flow exhibits an initially unsteady one-dimensional flow which transits to a steady two-dimensional flow with the passage of a signal travelling up the plate. The travelling signal is shown to consist of an initial rapidly travelling component localised in the region of the plate origin, followed by a slower travelling component. The velocity of the slower travelling component is shown to be well predicted by the maximum velocity travelling wave obtained from a stability analysis