Thursday, October 17, 2019

Working Pressure in Momentum Equation

Momentum equation in (Navier-Stokes equations) can be written as

Pressure in the above equation is the thermodynamic/mechanical (assuming Stokes hypothesis). But CFD codes in general don't use thermodynamic pressure in momentum equation discretization. They instead define a working pressure and use that. Here are the steps that take us to the final equation that is used for discretization.

Step 1

Introduce a constant reference density.


Introduce a constant reference pressure such that




  • Step 1 helps in decreasing the magnitude of gravity source term that can otherwise have some destabilizing effects. Step 2 is done to avoid precision issues. In general, the variation in pressure is orders of magnitude smaller than the mean value of pressure. If we work with absolute magnitude of pressure in discretization, we will waste most of the significant digits and may not capture variation in pressure to desired accuracy.
  • In a stagnant (constant density) liquid column, if we chose reference density to be liquid's density then p_therm will increase linearly as we go deeper but p_working will be constant.

Wednesday, October 16, 2019

Incompressible Flow

Incompressible Flow Definition

Continuity equation can be written as

If the material derivative of density is 0 (or negligible) i.e. if the density of a material particle does not change (or does not change significantly) as it moves through the flow domain, then the divergence of velocity vector is equal to 0 (or close to 0). Divergence free velocity is generally taken as the definition of incompressible "flow".

Note that constant density fluid implies divergence free velocity but the reverse is not true. Flows involving constant density fluid are always incompressible. Flows involving non-constant density fluid can also be assumed to be incompressible or divergence free under appropriate operating conditions. There is a difference between incompressible "fluid" and incompressible "flow" and that must always be kept in mind. Incompressible "fluid" implies incompressible "flow" but incompressible "flow" does not always imply incompressible "fluid".

Compressible / Incompressible "Fluid" and Incompressible "Flow"

For any fluid, equation of state (eos) can be prescribed to express density in terms of other thermodynamic variables - pressure and temperature. Fluids with dependence of density on pressure are called compressible "fluids" and those with no dependence of density on pressure are called incompressible "fluids". Strictly speaking, no fluid is incompressible. There is always some amount of compressibility, however small that may be. But for many fluids, it is fair to neglect compressibility for a wide class of problems. For example, water's density does not change appreciably when subjected to a wide range of pressures. For these fluids, we can simply neglect the dependence of density on pressure. And in many cases, we can go a step further and assume density to be constant.
  • Constant Density "Fluid" As mentioned above, any flow involving constant density incompressible "fluid" can be termed incompressible "flow" (i.e. divergence free velocity).
  • Ideal Gas "Fluid" At low Mach numbers, density variation in flows involving ideal gas can be very small. So the assumption of incompressible "flow" (i.e. divergence free velocity) is fairly accurate for low Mach number flows involving ideal gas.
  • Other Compressible "Fluids" For any general eos, we have to look for flow regime where the density variation over the solution domain is negligible. Over that flow regime, the assumption of incompressible "flow" would hold in general.

OK! Now we can identify incompressible "flow" based on eos and operating condition but how does it help?
  • Simplified Theoretical Analysis With the assumption of incompressible flow, simple formula can be derived that are easy and simple to use for theoretical analysis e.g. Bernoulli's equation and simple definitions of total/stagnation temperature & pressure.
  • Faster/Robust Solution Approach For a compressible "fluid", all the flow + energy variables (rho, p, V, T) are tightly coupled via the set of Euler (or Navier-Stokes) equations and equation of state. In general, this set of tightly coupled equations are solved numerically using what is generally termed as density-based approach. The approach has inherent coupling of both convective and acoustics velocity scales. At moderate and high Mach numbers, the two velocity scales are comparable but low Mach numbers the convective velocity scale is much smaller than acoustic velocity scale. This wide separate between convective and acoustic scales presents several problems with density-based approach. So people started looking for alternatives and pressure-based approach was derived with the assumption of incompressible flow. Pressure-based approach decouples the solution of momentum equation (based on convective scales) from that of pressure (based on acoustic scales) and provides a much better (faster & robust) alternative to density-based approach for incompressible flows. So if we have identified that the problem under consideration involves incompressible "flow", the choice of which numerical approach to use is simple.

Thursday, September 19, 2019

Why do we need Prism Mesh in Boundary Layer?

Let us consider a cell-centered unstructured finite-volume CFD solver. In the core of simulation domain, away from wall boundaries, we can use hex, tet, or poly cells.

Why can't we use the same type of cells in boundary layer? In the core of simulation domain, gradients are mild or negligible. But in boundary layer close to a wall boundary, solution gradients are extremely high normal to the wall but almost negligible along directions parallel to the wall. To resolve these highly anisotropic gradients we have two options to choose from:
  1. either use isotropic cells (similar cell size in all directions) with cell length scale based on the requirements of resolving wall normal gradient,
  2. or use highly stretched anisotropic cells with small length scale (based on the requirements of resolving wall normal gradient) in wall normal direction and much larger length scale in other directions.
Option #1 leads to a much larger number of cells compared to option #2. These extra cells, however, unnecessarily add to computational cost without improving accuracy in a noticeable way. That's why option #2 is preferred.

Can we use highly anisotropic hex, tet, or poly cells? Anisotropic tets and polys can have very bad skewness and non-orthogonality and solver has difficulty in handling them. (What do skewness & non-orthogonality mean and how is solver affected by them? We will leave that for a later post.) Anisotropic hexes, however, do not have the same problem. They can be highly squished along just one direction while maintaining zero skewness and full orthogonality. So
  • if the core mesh is hex, we use quad prims (stretched hexes - with quad bases) in boundary layer,
  • if the core mesh is tet, we use tri prisms (tri bases) in boundary layer, and
  • if the core mesh is poly, we use poly prisms (poly bases) in boundary layer.

Sunday, September 15, 2019

Brushy Creek Regional Trail

The full Brushy Creek Regional Trail (Cedar Park, TX) is around 10 miles long (one-way). Part of this trail extending from Brushy Creek Lake Park to Twin Lakes YMCA is around 6 miles round trip.

We cover parts of this trail often but today morning we did the whole 6 miles round trip. At a brisk pace it took us around 2 hours. This is a nice well-maintained concrete trail. Only the section near Brushy Creek Lake Park is paved. There are a lot of people walking up and down the trail over weekend mornings. Bikers share the same trail although there are several offroad shoots as well. The trail is mostly surrounded by trees on both sides. So even on a hot day you don't feel burnt down. The section close to Brushy Creek Lake Park, however, is directly exposed to the sun and can be difficult to walk after early mornings. Here are some photos from today's walk.

And here is a small video with soothing sound of flowing creek.

Saturday, March 30, 2019

Star Trek type Moon gif

Taken sometime in 2015 (or perhaps earlier) using Orion XT8i and Orion StarShoot Solar System Camera