# Friction Coefficient

Discussion in 'Hydrodynamics and Aerodynamics' started by jesdreamer, Sep 30, 2015.

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### philSweetSenior Member

Well, to start with, you are sorely abusing Mr. Bernoulli. What you can do with Bernoulli's equation is this. For a steady and incompressible flow, you know that the Total Head is constant along a steamline. So you can trade pressure and velocity off against one another keeping the Total Head the same everywhere along a streamline. So, to calculate the pressure at one point, you need the velocity at that point, and you need both the pressure and the velocity at some other point along that same streamline. Bottom line is you need to solve for the flow field around the object to do this. That is where the velocities come from.

The next problem is that following streamlines is not is an attractive way to get to a point on the hull where you want to know the pressure. The shorter path is to start away from the hull and cut across the streamlines to get to the hull. Obviously, the answer has to be the same either way. But you're not using Bernoulli's equation. Oddly, the equation doesn't have a name, but "streamline curvature theorem" is a translation from Japanese according to wiki. The change in pressure, dp/dr = rho*u^2/r, where r is the vector from the center of curvature to the streamline. The pressure is lower towards the center, greater on the outside of the curve.

In general, both equations are derived from Euler equations.

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### JoakimSenior Member

All the free-surface codes I know (panel method and full N-S with different kind of free-surface models) use the body surface pressure and thus method 1.

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### Mikko BrummerSenior Member

N_S codes for boat hulls are getting more and more accurate, as in Quequen's post in the Sysser 62 thread.

Attached my results for a rather ad-hoc, 2 DOF (trim & heave) study on another Delft hull, Sysser 50 (this wasn't N-S but Lattice Bolzmann, though). In the second attachment, results from Chalmer's, with a code called Shipflow, really for tankers & ships. FineMarine by Numeca is considered very good, as is Star-CCM+, both used by many yacht designers.

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### DCockeySenior Member

Bernoulli's equation and Euler equations are only valid for inviscid flow where viscous effects are insignificant. "Total Head is constant along a steamline" only if the flow is inviscid. "The change in pressure, dp/dr = rho*u^2/r, where r is the vector from the center of curvature to the streamline" is valid only if the flow is inviscid.

External flow around a body can be considered as invisicid sufficiently far from the boundary layers along the body surface, any separated flow regions, and viscous wake downstream of the body, provided the approaching flow has sufficiently low turbulence. If the approach flow is uniform with zero vorticity then the flow sufficiently far from the body will also be irrotational and the potential flow assumption can be used which leads to further simplification.

Invisicd flows can be solved for reliably using CFD and similar methods provided the boundary conditions are known for the inviscid flow region. Potential flows are generally simpler to solve for and singularity methods can be used.

The difficulties in solving for external flows arise in the regions where the flow is not inviscid. If the flow along the body is attached then the boundary layer assumptions can be used, and if the boundary layer is sufficiently thin then the pressure on the surface of the body can be assumed to be equal to the pressure just outside the boundary layer. However if the flow is separated then the boundary layer assumptions are not valid and solving for the flow is much more complex and difficult as the full Navier-Stokes equations are needed with appropriate turbulence modeling.

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### daiquiriEngineering and Design

Just as a short pedantric note, if you don't mind.

It is true that Bernoulli's equation has been formulated in its original form for inviscid fluids. However, there is also the so-called extended Bernoulli's equation, which is essentially an energy equation and contains viscous losses:
p1 + 0.5 rho V1^2 + rho g z1 = p2 + 0.5 rho V2^2 + rho g z2 + Qvisc​

where Qvisc is the dissipated energy per unit volume due to friction.

This formulation is the basis for calculations of air ducts and hydraulic piping (for example), where Qvisc is usually either tabulated for various types of local obstructions or given through graphs or equations (such as Darcy-Weisbach and Colebrook-White formulae) for distributed (wall friction) losses along the path.

Cheers

Last edited: Oct 2, 2015
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### Leo LazauskasSenior Member

Both methods are widely used.

Body surface pressures are very sensitive to the hull definition. Small
undulations in the surface (either real, or in the input form) can make a large
difference to estimates.

Other methods, like Michell's (thin-ship) wave resistance integral, can also
present difficulties. The original form of the equation was expressed in terms
of hull slopes. It's often better to use integration by parts to convert the
integral so that hull offsets can be used.

There are some (pathological) cases where pressure integration is so difficult,
that a far-field procedure is preferable. For example, a rectangular pressure
patch representing a hovercraft is quite easy to do with either method.
However, if the patch is oblique to the flow, it is better to calculate the
down-stream waves and then use Fourier methods to extract the wave
resistance.

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### markdrelaSenior Member

The I said "drag estimation", I meant drag estimation using empirical methods. I was not referring to NS codes.
I reworded that post to clarify.

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### DCockeySenior Member

And there is an equivalent version used by civil engineers for flow in rivers, over dams, etc. But neither is of direct use in solving for external flow around bodies.

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### daiquiriEngineering and Design

I agree. Perhaps it might be theoretically possible, but it is not being done in practice because of the impossibility to find a universally valid expression for the dissipation term in the formula, except for some simple mono-dimensional cases.

What I wanted to show in the previous example is that Bernoulli's equation can be re-written for the viscous case too, and that the viscous formulation is commonly used for solving certain practical engineering problems.

Cheers

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### markdrelaSenior Member

The main problem with applying your formula is that Qvisc is not definable in terms of flow quantities at the point in question. The formal expression for Qvisc is an integral of the transverse shear stress gradient d tau / dn along a streamline, all the way from far upstream:

rho Qvisc(s) = Int (d tau/dn) ds'

where the dummy variable of integration s' runs from -infinity to s. In general, you don't know what Qvisc is at any one point without knowing what happened along the streamline all the way upstream. This spoils the "endpoints-only" character of the Bernoulli equation.

Note also that Qvisc is not exactly the accumulated viscous energy dissipation, and doesn't even have the right units for dissipation, since it's missing a factor of velocity.

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### jesdreamerJunior Member

Back to Cave Man Concepts

Read my title for this specific post -- At the risk of alienating the experts, I feel discussion is getting into too much esoteric mumbo jumbo. To address DCockey's education question per post #14, I have had one university level fluid dynamics course, it was at RPI but in 1952 or 53 so I might not still have much grasp of fundamentals & virtually zero of details of formulas involved.

I seem to be able to follow PhilSweet's posts and per his #6 do want to go into basic drag concepts a little further than just friction. I'd like to go back to my post #8 to explore sinkage/trim effects and (if) any related component of drag force.

Joakim #9 seems to agree on an induced drag related to lift (aerofoil) but sees little (no?) trim in displacement hulls (thus no related induced drag?). I disagreed(#13) and cited dramatic squat & even swamping of canoes & kayaks at paddler speeds. If I am right I would like discussion on the induced drag due to hull squat. Several posts refer to friction parallel to flow and pressure perpendicular -- I get the impression posts mean perpendicular to distant water surface (but I feel the pressure forces are acting perpendicular to hull curved surface at any point in question, with a component parallel to water free surface and another component perpendicular to water free surface, am I right?). And if trim yields a drag force I would like a way to estimate it -- also I am interested in whether trim tends to balance out any pressure differentials which caused it in the first place (does trim lead to pressure equilibrium?).

Again I go back to PhilSweet (who I seem to understand) in #16 he gets into conservation, balancing pressure and velocity -- It seems to me that if we have an overall speed of say 5kts, then via simple geometry we should be able to calculate speed at any point along hull (I realize zero slip) and related pressure differential (I don't know how to get a "starting pressure" from which to calculate actual pressure at point in question).

I would like to calculate pressure at points along hull, perhaps as a way to determine trim (trim force) and thus to get to an estimate of induced drag (assuming hull squats until pressure equilibrium) -- If we can do this with some logic and simplicity, then I would like to know how we could determine approximately where separation should occur --

Or to look at this another way, if increase in trim might result in pressure equilibrium, might it also prolong flow attachment?? (with the trim induced drag force as the price of reaching this condition??)

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### daiquiriEngineering and Design

Oops, there is a typo in that extended Bernoulli's equation.
There should be no "rho" term in front of the Qvisc, the correct equation is:
p1 + 0.5 rho V1^2 + rho g z1 = p2 + 0.5 rho V2^2 + rho g z2 + Qvisc
where each single term has the unit of N/m^2 = kg/(m s^2) = J/m^3 (energy per unit volume).
The units are now correct and consistent with the integral equation
Qvisc(s) = Int (d tau/dn) ds'
I will modify the post #20 too, in order to not mislead someone. It was my fault, thanks for noting it.
Cheers

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### DCockeySenior Member

"Pressure perpendicular" in the context it has been used here refers to pressure as the local force element acting perpendicular to the surface. The local force field from surrounding fluid on any object can be split into the pressure field which acts perpendicular to the surface and the shear/"drag" force which acts tangential to the surface. That is both a result of the physics as well as the technical definitions.

A boat will sink/trim/heel until the overall forces acting on the boat, including gravity and the integrated pressures and sheer/drag stresses, are in equilibrium. However the pressure at any point will generally be different than if the boat was restrained for sinking/heeling/trimming.

If the flow around the body is attached and an inviscid assumption can be made then as I noted in a previous post the calculations are simplified compared to the extremely complex problem of solving for viscous flow, but it is still far from a matter of simple geometry because the velocity at any point on/near the surface of the body is affected by the overall shape of the body.

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### philSweetSenior Member

Me too.

This isn't induced drag. Lets just call it added drag due to a change in trim.

This is a hard problem. Realistically, this is where tank test data excel. You can input the sinkage and trim from a tank test and forgo having to float your mathmatical model. This allows for much simpler, faster models using the far field calculations for drag, which, unfortunately, can not be directly coupled to sink and trim forces. (Leo, a little help here please.)

It helps to get a feel for the magnitude of the forces you are looking for.

Lets say the total drag on a boat at sea is 5% of it's weight. This is what the propulsion system is producing.

Maybe 3% points are due flatwater drag with a clean hull (which you want to model), and 2% points due to fouling and seaway and air (which you are choosing to ignore).

The 3% points can be divided into stuff that scales with the Reynolds number, and stuff that scales with Froude number. Lets say 1% point and 2% points respectively. Both influence sinkage and trim.

If the local free surface that the boat is floating on is tilted bow up, it doesn't need to be by much for the pressure vector's drag component to be changed by an amount of the same size as the other components. This is while holding the trim as measured at the local free surface constant. Then there is trim that affects the immersed hull shape, ie a change in cg. or an applied moment. The change in drag is different in this context.

And then there is sinkage. If there is sinkage, a round-bellied boat is floating in a hole of its own creation and there is a pile of water somewhere nearby. If the pile(s) and the hole are not symmetric, then there must be a net force acting laterally (such as drag). If there is a rise in cg, then the boat is floating on the pile and there is a hole somewhere nearby. In planing, there isn't much of a pile of water or hole nearby, there is just a boat-sized hole of displaced water, however, the boat is only partially in it.

It is the pressure profile on the hull of the boat (which is a function of the physical hull, its orientation, any boundary walls or floors, the hull motion, the fluids, and gravity) and the pressure profile on any boundaries, that determine the free surface everywhere (isentropic). And the free surface close to the boat has a great influence on the details of the drag vs velocity plot.

Coupling the curvature of the free surface and pressures on the hull and the orientation of the free floating hull is hard. There are simple models which have the charming characteristic that the same set of assumptions are made for the various pieces of the model. They are consistent mathematically and usually not very accurate wrt trim and sinkage. There are better models that mostly have inconsistent assumptions, but may gain more in precision than they loose in accuracy, at least with respect to selected variables such as trim. Or you can go whole hog with Navier Stokes, but the electrical bill for the computation may run more than the cost of the hull.

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### jesdreamerJunior Member

Trim "Related" Drag

PhilSweet in post #29 states the following -- "If the local free surface that the boat is floating on is tilted bow up, it doesn't need to be by much for the pressure vector's drag component to be changed by an amount of the same size as the other components...."

When I raised the question of trim effect on drag I was visualizing a kayak with 100# added to rear deck and the suspicion that this might yield a 1 or 2 degree angle of attack -- I realize this load would still be acting vertically downward but could visualize that if a trim of this magnitude was to take effect per Bernouli forces, then the drag component might be larger than total of all the other resistance forces combined -- so the situation might well justify consideration --

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