Hi, Does anybody have experience on simulating the flow across a small fan (used in summer)? Should the flow be simulated as incompressible? If so, is there any pressure increase after the fan? Thanks.

I do not have any experience with this kind of simulation. However, I believe I can still answer your two questions, by mentally visualizing the flow.

Yes, the flow is very low Mach number, and therefore can be treated very accurately as being incompressible. And yes, the air in general experiences increases in both kinetic energy and pressure as it passes through the fan. This is different from the simpler situation of inviscid incompressible flow through a duct without fan, where the Bernouilli equation says that in order for the kinetic energy of the air to increase, the pressure must decrease. In the case you are considering, the air is indeed flowing incompressibly, and the flow pattern resembles that in a duct (you can imagine a stream surface with a circular crosssection as representing the wall of a duct). However, here the fan acts as an external force on the air. If you are not modeling the details of the fan blade motion, at the very least you have to make the presence of the fan felt as a magical body force (momentum source) in your momentum equation, localized at the axial position of the fan. The action of the fan results in a pressure immediately upstream of the fan which is lower than the ambient pressure, causing the upstream ambient air to accelerate toward the fan. The fan also causes the pressure immediately downstream of the fan to be higher than the ambient pressure, which causes the air that has just passed through the fan to accelerate further in the axial direction. In the 2D or 3D cases, further downstream of the fan, the airstream (jet) diverges because of the lower pressure on the sides of the jet, and the air recirculates back to the upstream side of the fan while bypassing the fan via the side. I believe that a quasi-1D simulation is not as appropriate, since here the air would have no chance to return upstream by bypassing the fan, and this would therefore change (from ambient) the pressure at downstream infinity (or would change the velocity (from zero) at downstream infinity).

Thanks a lot!

I think Ananda suggested some very good points regarding flow simulation of a fan. But I have the following concerns: Since the flow is incompressible, the density will not change. Suppose the area of the duct is constant, continuity equation requires that the velocity remains the same after the fan (a single stage rotor) assuming there is no swirl. According to P=rho*R*T, the temperature will increase after the fan. For the flow from the plane after the fan to the exit of the duct, Bernoulli can be used and I think since the pressure after the fan is higher than ambient, there will be a flow acceleration after the fan in the duct to match the atmospheric pressure boundary condition in the exit. It's kinda confusing because I always think that temperature does not come into consideration for incompressible flow.

You are quite welcome. I should add that the simplest model for your case is that of a doublet. The flow near your fan, but not within the fan location itself, is similar to the 3D flow induced by a doublet. The axis of the doublet is the axis of the fan. The strength of the doublet is related to the volume flow rate through the fan. If you remember the equations for doublet flow (without superposing a stream uniform at infinity), or look them up in any good book on fluid dynamics, you will be able to estimate velocities and pressures in the fan nearfield with paper and calculator, without even using a CFD code. However, if your purpose is to improve the blade design of the fan, or to predict the acoustic noise made by the fan, you must resort to more detailed modeling of the blades, etc.

The fan is also imparting considerable angular momentum to the air passing through it. The interplay between the angular velocity and radial velocity shows up in the Coriolis and centrifugal 'forces' in those two momentum equations.

I think that's true for real compressor, but for a very small fan (used in summer to get some cool air), the swirl might not be important.

My previous posts assumed a small fan of the table or floor-standing variety, operating in a closed room much larger than the fan. Now that you mention a fan in a constant-area duct, I visualize a fan, perhaps in a duct set in a room wall, exhausting to atmosphere outside the building. The flowfield is rather different than in the previous case, but the action of the fan is similar. For the sake of definiteness, let us assume a constant-area duct.

First, let us consider inviscid incompressible flow through a constant area duct with no tip clearance of the fan. The air will have constant axial velocity throughout the duct (because of the continuity equation). The pressure in the duct all the way upstream of the fan will be uniform and lower than room pressure. The pressure in the duct all the way downstream of the fan will be uniform and higher than the atmospheric pressure that the duct is exhausting to. The flow in the room in the vicinity of the duct inlet will resemble the incompressible flow in a rapidly converging nozzle. However, because of viscous effects (boundary layer separation), the flow outside the duct exhaust will not be a reflection of the flow ahead of the inlet. Rather, the flow will issue from the exhaust as a jet, and then the jet will actually contract in diameter until the pressure in the jet falls to match atmospheric pressure. Thereafter, if viscosity is ignored, the jet might continue with constant diameter all the way to infinity, but in reality viscous shear layer mixing, usually turbulent, will convert the kinetic energy of the jet to internal energy. Far away from the exhaust, the jet will therefore slow to a standstill and become warmer. While it is slowing down it will also undergo lateral spreading. Thus the flow pattern after the duct exhaust will be much more complex than the pattern before the duct inlet.

In steady, adiabatic flow, both compressible and incompressible, the stagnation enthalpy is constant along a streamline upstream of the fan, and along a streamline downstream of the fan. At the fan itself, the stagnation enthalpy undergoes a rather rapid increase due to the constant-volume compression work done by the fan. In steady truly incompressible flow, following a streamline upstream of the duct inlet, the pressure is decreasing while the velocity is increasing. Similarly, along a streamline IMMEDIATELY downstream of the duct exhaust, the pressure is decreasing while the velocity is increasing. These are redistributions of the stagnation enthalpy according to the Bernouilli equation. Still further downsteam, after the vena contracta, in a real viscous flow, the stagnation enthalpy is further redistributed, with the kinetic energy being converted this time to internal energy (this does not obey the Bernouilli equation, but the First Law of Thermodynamics for Steady Flow is still obeyed). Within the constant-area duct, the axial velocity is constant all through. In the duct portion upstream of the fan, the pressure and velocity are constant. In the duct portion downstream of the fan the pressure and total velocity are constants differing from their upstream values. At the fan itself, corresponding to the rise in stagnation enthalpy, there is no change in axial velocity, but rather a steep increase in the pressure (the difference between the pressure side and the suction side of the fan blades). Of course, if swirl imparted by the fan is taken into account, some of the stagnation enthalpy rise across the fan appears as a pressure rise, and some of it appears as kinetic energy of downstream swirl, with no change in the kinetic energy of axial flow because of the continuity equation.

Next, regarding the incompressible limit of compressible flow. In the limit of zero Mach number (infinite speed of sound, modeled by letting gamma of a compressible gas tend to infinity), the energy equation (and hence any temperature effects) gets precisely decoupled from continuity and momentum equations, though the decoupling is not mutual. One can solve first for velocity and pressure, and then solve for the temperature from the energy equation after the fact. In the incompressible limit, with an inviscid non-conducting fluid, and no heat sources/sinks such as chemical reactions, the temperature field simply gets convected. The temperature could have a wide spatial variation at any instant. Bear in mind that the perfect-gas equation of state is invalid for truly incompressible flow.

In slightly compressible flow of a perfect gas, in flowing through the fan, the air undergoes increases in both density and temperature, and an even larger fractional increase in the pressure. In the regions outside the duct inlet and exhaust, the air undergoes decreases in both density and temperature whilst undergoing larger decreases in the pressure. Neglecting the effects of viscosity, these processes may be modeled as being isentropic. You can work out for yourself from the isentropic expansion/compression relations what the changes in density and temperature are like relative to the change in pressure. Anyway, it will turn out that provided the pressure ratio across the fan is not large (true for a ventilation fan), the pressures and velocities seen in an incompressible flow calculation are pretty close to those in a calculation at the true (low) Mach number. The densities and temperatures of the two flows will of course differ.

A better duct design might be a converging or converging-diverging one. In this case the acceleration and deceleration take place partially inside the duct. If significant rounding of the duct inlet and exhaust is not done, there will be inefficiencies in moving the air (more of the change in stagnation enthalpy imparted by the fan will go toward heating the air than moving it). Also, accounting for the tip leakage (spillage), which occurs in any real fan, shows that some of the flow recycling occurs that I mentioned in an earlier post. I recommend axisymmetric moderate-Re Navier-Stokes incompressible or low-Ma simulation at the very least.

Hope that this rather long-winded explanation helps clear up your confusion rather than adding to it.

Jim Park is right. The doublet is a crude model, and lacks a lot of the features of the real flow. It ignores the swirl, and it also ignores the work to be done by the fan. It also ignores the fact that in real life, due to boundary layer separation, the flow after and ahead of the fan are not symmetric mirror images of each other. See my rather lengthy post in reply to Jane's further doubts.

I forgot to clarify that in the incompressible case, the pressure rises across the fan, but the density and temperature remain constant. The truly incompressible fluid does NOT obey the perfect gas equation of state. Further downstream of the contracta vena in the exhaust jet, the temperature rises due to viscous dissipation.

Thanks a lot for Ananda's message, it's really helpful. What if there is some heat transfer between the wall and the fluids downstream of the fan? In this case, can the flow still be simulated as incompressible? Is mach number the only criteria to judge whether the flow is compressible or not? Many thanks.

A viscous, heat-conducting fluid in incompressible flow can still model the heat transfer from the wall very well for this case. The Mach number is indeed the principal criterion to judge the compressibility of the flow. The main reason for modeling a low-Mach number flow as an incompressible flow is that you can reduce the computational effort, particularly in the case that heat transfer is unimportant and you don't care about the temperature field. In the latter case, you have one less equation (the energy equation can be dropped) and one less variable (the temperature) to carry. Even in the case where heat transfer and temperature variations are important, because of the decoupling in incompressible flow, you can solve a smaller system (continuity and momentum) first for velocity and pressure, then the energy equation for temperature. Also, with compressible flow solvers, you generally have to do something special with the numerical algorithm to make it work well at low Mach number (preconditioning, implicit schemes).

However, these days, I would say the computational efficiency is less of a concern, and I myself would prefer to use a suitably preconditioned or implicit compressible flow code and run it at the correct (low) Mach number. I don't think it costs that much extra to run a compressible flow code versus an incompressible flow one. It all comes down to whether you have a good compressible flow code available that can handle low Mach numbers, or whether you have a good incompressible flow code available. For Mach numbers lower than about 0.1, it makes little difference which approach you use.

As I tried to point out, the low Mach number limit is not a singular limit, and slightly compressible flow is a regular perturbation of incompressible flow, not a singular perturbation. One can either carry out a compressible flow simulation at the right Mach number, or alternatively carry out an incompressible flow simulation. For higher accuracy in the latter case, some kind of post-processing correction of the velocity and pressure fields could be done based on the square of the Mach number. See the subsonic similarity rules chapter of a good fluid dynamics text.

Glad I could be of some help.

I think you should search for papers and books about classical theories for propeller, wind turbines and fans such as momentum theory, blade element theory, vortex method and vortex lattice method.

Have a check on Fluent if you want to use CFD commercial software to do a simulation.

Regards

Li

Many thanks to Ananda. Right now I have a much clear understanding of what's going on for the flow in a duct fan.

A ducted fan is very different from a fan without a duct. The duct plays an important role in the efficiency of the fan.

Li

Li Yang mentioned Fluent's work here. Now that the physics has been explained, one needs to properly section the problem so that each region can be conveniently solved: http://www.fluent.com/solutions/articles/ja118.pdf

The other half of the effort is to transfer the information from one reference frame to another seamlessly...the specifics of which I'm sure are proprietary.