3: Flow Past a Sphere II - Stokes' Law, The Bernoulli Equation, Turbulence, Boundary Layers, Flow Separation - Geosciences
- 3.1: Introduction and the Navier-Stokes Equation
- An equation of motion for a viscous fluid when the forces acting in or on the fluid are those of viscosity, gravity, and pressure, is called the Navier–Stokes equation. The Navier–Stokes equation is notoriously difficult to solve in a given flow problem to obtain spatial distributions of velocities and pressures and shear stresses. Basically the reasons are that the acceleration term is nonlinear, meaning that it involves products of partial derivatives, and the viscous-force term contains secon
- 3.2: Flow Past a Sphere at Low Reynolds Numbers
- We will make a start on the flow patterns and fluid forces associated with flow of a viscous fluid past a sphere by restricting consideration to low Reynolds numbers.
- 3.3: Inviscid Flow
- Over the past hundred and fifty years a vast body of mathematical analysis has been devoted to a kind of fluid that exists only in the imagination: an inviscid fluid, in which no viscous forces act. This fiction (in reality there is no such thing as an inviscid fluid) allows a level of mathematical progress not possible for viscous flows, because the viscous-force term in the Navier–Stokes equation disappears, and the equation becomes more tractable.
- 3.4: The Bernoulli Equation
- In the example of inviscid flow past a sphere described in the preceding section, the pressure is high at points where the velocity is low, and vice versa. It is not difficult to derive an equation, called the Bernoulli equation, that accounts for this relationship.
- 3.5: Turbulence
- Most of the fluid flows of interest in science, technology, and everyday life are turbulent flows—although there are many important exceptions to that generalization. Turbulence might be loosely defined as an irregular or random or statistical component of motion that under certain conditions becomes superimposed on the mean or overall motion of a fluid when that fluid flows past a solid surface or past an adjacent stream of the same fluid with different velocity.
- 3.6: Boundary Layers
- A boundary layer is the zone of flow in the immediate vicinity of a solid surface or boundary in which the motion of the fluid is affected by the frictional resistance exerted by the boundary. The no-slip condition requires that the velocity of fluid in direct contact with solid boundary be exactly the same as the velocity of the boundary; the boundary layer is the region of fluid next to the boundary across which the velocity of the fluid grades to that of the unaffected part of the flow.
- 3.7: Flow Separation
- 3.8: Flow Past a Sphere at High Reynolds Numbers
- 3.9: Settling of Spheres
- This section deals with some basic ideas about settling of solid spheres under their own weight through still fluids. This is an important topic in meteorology (hailstones), sedimentology (sediment grains), and technology (cannon balls and spacecraft). In this section we will look at the terminal settling velocity of spheres as an applied problem. At the end I will make some comments about the complicated matter of the time and distance it takes for a sphere to attain its terminal settling veloc
In this section, let us present a number of “cookbooks” – examples of how to use ASPECT in typical or less typical ways. As discussed in Sections [sec:running] and [sec:parameters], ASPECT is driven by run-time parameter files, and so setting up a particular situation primarily comes down to creating a parameter file that has the right entries. Thus, the subsections below will discuss in detail what parameters to set and to what values. Note that parameter files need not specify all parameters – of which there is a bewildering number – but only those that are relevant to the particular situation we would like to model. All parameters not listed explicitly in the input file are simply left at their default value (the default values are also documented in Section [sec:parameters]).
Of course, there are situations where what you want to do is not covered by the models already implemented. Specifically, you may want to try a different geometry, a different material or gravity model, or different boundary conditions. In such cases, you will need to implement these extensions in the actual source code. Section [sec:extending] provides information on how to do that.
The remainder of this section shows a number of applications of ASPECT . They are grouped into three categories: Simple setups of examples that show thermal convection (Section [sec:cookbooks-simple]), setups that try to model geophysical situations (Section [sec:cookbooks-geophysical]) and setups that are used to benchmark ASPECT to ensure correctness or to test accuracy of our solvers (Section [sec:cookbooks-benchmarks]). Before we get there, however, we will review how one usually approaches setting up computations in Section [sec:cookbooks-overview].