What we need to know to determine the flow evolution

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Suppose that we know the velocity and pressure distribution at a certain time instant and in the entire space. Can we decide, intuitively, what the velocity field will be like a little time later? The answer is yes.  First, since we know the velocity, we can figure out the displacement of the fluid particles. Without pressure and viscosity, advection alone (that is –u ) would result in the velocity remaining the same at each particle. Hence, the velocity at a point in space will change accordingly, if it was different at the nearby points. The pressure will add to this change. The pressure contribution will be directed opposite to the pressure gradient (∇p), that is it will be a vector pointing in the direction of the fastest decrease in pressure, with the length proportional to the rate of this decrease. Viscosity, too, will contribute, leading to diffusion of the velocity: if the velocity at a point is below the average of the velocities around it, the velocity at this point will increase due to diffusion. The higher is the value of Re, the smaller this effect will be.

Now, knowing the velocity a little time later, we can repeat this reasoning again and again, stepping in time, thus finding the flow at all later times in a time-marching procedure. However, for this we also need to know the pressure. Unfortunately, there is no intuitive way to determine the time derivative, or, which the same, the rate with which pressure varies with time. The physics determining the pressure evolution is complicated, at least in the general case. To understand it we have to consider the property of incompressibility of the fluid together with mass conservation. This property, expressed as ∇·u =0 in the page devoted to this, is the property of the velocity field. It happens that the advection of velocity (the term –u), acting alone, might result in the new velocity field, which does not have this property, and for an incompressible fluid this violates the mass conservation law. The pressure corrects this. For a given velocity distribution at a certain instant the pressure distribution will be such that at the following instant the velocity distribution continues to satisfy mass conservation for incompressible fluid.  (This is hard to use intuitively, but this difficulty will be resolved in the following pages.) The important observation is that to start the time-marching procedure one does not need to prescribe the initial distribution of pressure: the initial velocity field is enough.

If the fluid fills the entire space then nothing else is required. However, if the region occupied by fluid has boundaries then for time-marching one also needs to know what happens at those boundaries, namely, to know what the velocity is at the boundary at all times. This cannot be determined from time-marching. Instead, it has to be prescribed.

The most common boundary is the surface of a solid body. The fluid cannot penetrate the surface of the body, which limits what the fluid velocity at the surface can be. Moreover, due to friction between the surface and the fluid, at the surface the fluid velocity is simply equal to the velocity of the solid surface. In particular, if the body does not move (say, a building in a wind), then at the surface the fluid velocity is zero.

Another common case of a boundary is an imaginary boundary, which we add simply because we know what the velocity at that boundary is. For example, for the case of an aircraft flying in an otherwise still air, we can imagine a box around the aircraft, so large that on the surface of the box the air velocity is zero. Then we will not need to think about the fluid motion outside the box, which makes it easier.

The summary of this page: the fluid motion is defined by the initial conditions for velocity, and the boundary conditions for velocity. No initial or boundary conditions for pressure are required.

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