just the drag of that fin, please
I am curious what the force that hold me back at such speed ( board resistance we could skip for now).
I guess that fin manufacturer doesn't bother about computing n-s , just attach sensor on the board and run at testing pool at speed required, Support frame run above water or rails by pool side.
Result is instant and correct in real life, but computer model could be more interesting since allow to develop the "perfect fin", not by a chance but by the design.
I guess Mythbastards could also attach the B42 hand ( obviously made of jelly) to that testing frame to check if 350kg and 75 km/h will brake that hand
Hi I
So, just to summarise what you said (correct me if I'm wrong).
The equation expresses a bunch of particles, but lets just look at one particle. Here is an expanded left side of the equation (sorry about my crappy pics).
The left side of the equation expands to two terms.
The first term is force (mass times acceleration) that most of us are familiar with. The 2nd term converts the force (the 1st term) from a relative reference system to an absolute reference system.
If a boat is going down a river. If I'm on the boat, my view is a relative view point (relative reference system). If I am standing on the bank of the river, then my view point is an absolute view point (absolute reference system).
I must admit Ian. My knowledge of vector geometry is basic at best. I know how they work in pictures, but I certainly don't use them day-to-day. So I'm not going to bother agonising over the 2nd term.
Okay, that makes things a bit clear with the left side, but what about the right side of the equation ?. We have a 'p', a 'mu' (ie: a funny 'u' character), a 'v' (velocity ?), and a 'f'. We have 3 terms, and more vector geometry !. Could you talk us through the right side of the equation Ian ? (one little particle, not an entire field) :)
J
You're getting there Jn1, I remember spending a lot of time trying to get the n-s equation on board. It's worth the battle, it's only f=ma for fluids.
Getting the system of reference sorted is the first step. It helps to look ahead to how this equation will be used in a computer because that is why it is expressed like it is.
If you were modeling the flow around a fin for instance you'd set up a grid of a whole lot of little boxes around the fin. As the computer program runs each little box will have two lots of data.
the pressure of the fluid in that box
the 3d velocity vector of fluid passing through that box.
For boxes that happen to be at the surface of the fin you'd constrain the velocity to zero. You'd let the computer fiddle with pressure in each of these boxes, because of course the pressure distribution over the surface of the fin is what we want.
Now back to the first term on the left hand side. The first term is the partial time derivative of the 3d velocity vector at a fixed location. ie in each box, how much does the velocity vary in the next time step of this computer program? For steady state flow the velocity will remain unchanged, and for modeling steady state flows, such as the pressure distribution on the fin at a steady 40 knots, this term will be zero. (It may be non-zero before the program settled down to give the answer, although computer programmers may have tricks to get the flow up to speed without worrying about this. Not sure)
So for steady state flow all the acceleration as the fluid particle moves is given by that 2nd term in the brackets on the LHS. The partial derivatives of velocity with respect to distance are of course easy for the computer to calculate just by looking at the neighbouring boxes.
The first term on the right is the gradient of pressure. Looking at our computer program that's making sure all the values in the boxes add up according to the n=s equation we can see if the pressure on one side of a grid box is greater than that on the other side of the the box there is a force on the fluid particle that is passing through the box at this particular instant of computer time.
The 2nd term on the right is the viscous force. The mu is the viscosity coefficient, honey vs water. If the fluid above is sliding forward more than the fluid below is dragging there will be a viscous force acting on the fluid in the box. It's the 2nd derivative because the first derivative. ie sliding forward above = drag back below it cancels out. The computer can work out the second derivative of the velocity components quite easily just by looking at the velocity data in neighboring boxes. This is where the Newtonian assumption comes in, shear stress is linearly proportional to velocity gradient.
The last force is the body force gravity, or something electrostatic, call it zero.
This view is all based on a computer doing what's called a finite difference method. You let the computer step through time in very small fractions of a second. At each time step it looks at each box in the grid and lets it interact with its neighbors according to the n-s equation. The computer should settle down and give the stable smooth flow. Mostly it crashes. These days there are more sophisticated, less intuitive ways of solving this equation in a computer
Although the Equation is for incompressible flow, incompressible flow is not ensured by the equation. There's an extra equation to look after this.
Correct me if I'm wrong Ian. Here goes:
The left side of the equation describes a particle (I'll talk about one particle for simplicity). The ∂v/∂t term describes the acceleration of the particle in absolute reference, and right term converts it to relative reference (so it's now from the point of view of the particle's last position in time).
Looking at the right side of the equation, the first term is the pressure (force) pushing against the particle (that is described in the left hand side of the equation) as it moves through a viscous-"less" fluid.
Note: The negative sign in the 1st term tells us it's an "against" force.
The 2nd term (viscous force) is a transformer. It converts the force caused by the viscous-less fluid described in the 1st term, to a force of a viscous fluid (as set by the viscosity coefficient - μ).
The last term is the force (inertia ?) of the particle that opposes the force described in the 1st term.
So in summary, the left side of the equation deals with movement (velocity) of a particle caused by the force acting on it. This force is described by the right hand side of the equation.
Am I in the ball park ? . If not, then I had better stick to electronics.
J
We're probably the only two still tuned into this thread jn1 At leat we're not getting red thumbs.
I might leave it there Ian. I think I have the gist of it and that's all I need to know . I lack the experience in this field to fully understand the equation. I personally would need to experiment/play/observe before I even bothered looking at the maths (high and low pressure etc has lost me a bit).