Hydrodynamics


#21

FREE FLOW

Definition:

MODEL is a mental construction, a set of ideas that help for making mental images of phenomenons, making them easier to be understood.

Free flow happens when the motion of a fluid is not disturbed. Only the internal forces inside the fluid body have a role in the motion distribution of the fluid inside the studied volume.

For the understanding of many concepts, related to fluid dynamics, a moving fluid can be mentally modeled as a joint of layers that can slide one over the others. Imagine something similar to a pile of paper sheets.

Lets suppose a fluid in motion through a channel. The channel dimensions are so big, that the influence of the walls on the central fluid body can be neglected.

Lets forget, by now, why the fluid is moving.

When a layer moves, it drags the layer above and below, and both of them do the same with their sorrounding layers, resulting in a fluid where all layers are moving togheder with the same speed and direction.

If a line is drawn, representing all points of each layer, that are moving with the same speed, a set of parallel lines would be obtained. Those are the stream lines.

Stream lines cannot cross other stream lines. It would mean that those imaginary layers (the paper sheets) are crossing each other.

Lets now suppose that a sand grain is set on one of the layers. If the motion of that grain is followed along time, and a line is drawn, joining all points through which the grain has crossed, we'd get a path line.

Many times, a path line, and a stream line, can be represented by the same draw, but not allways. Will see it later.

How is the motion of a layer transmited to the others ? Just in the same way that it happens in a papers pile, by means of "INTERNAL" dragg.

The difference is that while inside a paper pile, each sheet is a seprated entity, in a fluid, the layers are only a mental model. The whole fluid body is an unique entity.

But its true, and can be observed, that even being a simple whole entity, a fluid whorks as if builded by those layers. There is an internal force that prevent any fluid part from moving with respect to the others.

This internal force is Known as Viscosity. The higher the viscosity, the more diffcult will be to distort the fluid. Viscosity works as a kind of flexible glue.

Imagine an alcohol drop falling along a tilted glass. It can be seen that the drop seems to be attracted by the glass. Having a close view to the drop, it can be realized that the drop shape is distorted by that attraction. In fact, if the tilting angle of the glass is reduced, there will be a given angle, before arriving to the horizontal, when the drop will stop moving. Gravity and that pseudo attraction are balanced.

If the same thing is done with oil, the drop will move slower, and the balance angle will be bigger. It looks that its harder to pour the oil than the alcohol.

This internal force that tries to stuck all parts of the fluid, and the fluid to everything around it, is called VISCOSITY.

The higher the viscosity, the closer will all layers move, becouse the force betwen them is higher.

Why have I chose alcohol and not water ? When a water dop is poured on a glass, other forces are taking part in the motion. Water, in that context, is a very special material.

Lets now approach the bottom of our huge channel, and lets have a look to the fluid layers very, very close to it.

The first layer, that which is in contact with the channel, will be stopped, or almost completly at rest. For all practical applications it is at rest.

The "second" layer is in contacyt with fluid, and not with the walls. A force, that we have not checked by now, is moving the fluid. And another force, viscosity, stucks this layer to the one below. Hence, this second layer will slide over the first, with a speed that will depend on the viscosity and the driving force(The one that is making the fluid flow).

Next layer will be sliding on an already moving layer, and hence, will move faster, and so on, until arriving to a layer where the speed is not affected by the walls. This is the free flow zone.

The higer the viscosity, the bigger the contact area beween fluid and walls, and the bigger any other force, stucking the fluid on the wall, the higher the dragg, and the thicker the affected area.

Lets now have a look to the driving force. In fluids, this driving forces do have no direct effect by themselves. Forces lead to pressure diferences, and fluid moves from higher to lower pressure.

Two areas with quite big pressure differences, will lead to fast motions. If the same areas with the same pressures, are moved appart, to a longer distance, the motion will be slower.

What really minds in a fluid motion is neither pressures nor distances, but the pressure diference that exists along a given distance.

This relation is called the "PRESSURE GRADIENT".

Lets follow the fluid along a given distance, from a high pressure area to a low one.

Along that path, driving pressure will be decreasing, from the higher to the lower pressure values. It can be said that the driving pressure at any point along the path, has a value that is located between the higher and lower values.

This pressure, that corresponds to intermediate points of the fluid along a path, is named "PRESSURE POTENTIAL".

Summarizing:

Imagine the fluid in 2D. As a vertical cut through the middle of the channel.

Lets take a very tiny squared portion of the fluid, and lets suppose it as a free body. Whats is the same, isolated from the rest of fluid.

Lets now try to imagine forces, in such a way, that the portion is balanced, and moving as it did when in the fluid.

We have four faces, and hence four forces are required.

For the vertical balance, buoyancy and weight are balanced. One point upwards and the other downward. There is no vertical motion.

For the horizontal motion: Even in that tiny portion, there is a distance beween the right and the left faces, and hence there the pressure potential at each one is different. One pressure points to the left, and the other to the right.

If pressure is different, the square will move from the higher pressure to the lower one.

Lets say it moves to the right.

But there is another force left, that has not been taken into account above.

The upper face, would be in contact with another fluid layer, that would eventually move faster than the square, that belongs to an intermediate layer. The lower face of the square, would be in contact with the layer below it, and hence, there must be two friction forces working, one at the upper and another at the lower face.

Both forces work horizontally, against the speed, but, are not equal. As it was said above, the lower one (in our example) is bigger than the upper. The result is that the square will be distorted. A shear will develop between both faces. This shear stress is another measurement of viscosity.

It must be realized, that all forces are exerted by the fluid body on itself. A portion of fluid influences other portions. That's why, those forces are named. INTERNAL FORCES.

If the explanation of our square, is extended to 3D, turning the square into a cube, and at the same time, the cubic sample is extended to the whole fluid volume under study, we'd arrive to the most commonly used fluid model.

Now, the thing will go around approaching how to guess te behaviour of each of those cubes, when the flow is distorted by any mean.


#22

ERRATA:

Why have I chose (choosen) alcohol and not water

There is an internal force that prevent (prevents) any fluid part.....

One pressure points to the left, and the other to the right. (Pressure potential points, not pressure).

We have four faces, and hence four forces are required. (We have four faces, and SIX forces working)

NOTE:

NEXT: Fluid properties and their relations.


#23

Well, in order to be a little bit less abstract, lets stop a little bit the general background, and lets have a look to some more practical questions.

In former posts, I've spoken about the behaviour of a fluid that flows around an object, about drag and lift forces ..... etc.

But, How is the real calculation procedure approached ?

The general solution for almost all fluid dynamics questions was proposed by Navier and Stokes. Those equations (quite, quite complex), take into account all variables playing in the fluids study. But being so complete, means to be very complex and really hard to be solved. In fact, there hasn't been already found a complete and stable solution to all of them.

For most practical applications simplified models, more or less complex, are applied.

One of the fundamental laws of the fluids Dynamics, is the Bernoulli relation, that states that Pressure and Speed are intimately related.

Suppose a tank, filled with water to a height of 10 meters. At the tank bottom there are a speed regulator and a pressure gauge that are connected to a pipe. Lets give that pipe a diameter of 0.05 meters (5 centimeters= 5/100 meters).

Lets neglect all kinds of turbulence, dragg, etc. Lets suppose a "perfect" system.

1-] The speed regulator is closed. Fluid speed, of course is zero. But ... what pressure will the gauge be measuring ?
As nothing is moving, the pressure will be that, exerted by the water filling the tank. The Hydrostatic pressure alone.

2-] The speed regulator fully opened. Fluid speed is maximum. And once again .... what will be the pressure measured by the manometer ?
Afree flowing fluid, does not exert any pressure on the pipe walls, its free for running. Hence, the manometer would measure zero.

In order for a fluid to exert any pressure on the walls of the pipe through which it is running, its speed must be reduced. Something must work as a fluid speed "brake". When the fluid is not allowed for running at its free flow speed, that brake force is translated to the walls, and "felt" as pressure.

3-] The speed regulator is opened a From 0% to 100%. Fluid speed will be that given by the hydrosstatic pressure, but reduced to a percentage given by the regulator.

What happens to the manometer reading ?
With the speed at 0, we read pressure is maximum and equal to the Hydrostatic pressure.
With the speed at 100%, we also saw that pressure is minimum and equal to zero.


Then between both values, pressure must change as speed changes.

While speed increases, pressure decreases.
Speed goes from zero to maximum, while pressure changes from maximum to zero. Hence ...

If a fluid accelerates, pressure will drop. If a fluid slows down, pressure will rise.


Bernoulli was the one that found the relation between both things. An stated that:

Pressure + 0.5 density Speed²= constant. With density= fluid density.
P + 1/2 density V²=K

Lets put numbers .....

Hydrostatic pressure is given by depth, and it's value for our example is:
Density: 1000 Kg/m³
Height: 10m
Gravitational constant(g)=9.81 m/s²

P= dens g H = 98100 Pa(Pascals)

1-] Regulator closed: Speed =0; Pressure= 98100 Pa.
From Bernoulli: P + 1/2 density V²= K.
K= 98100 + 1/2 1000 0²= 98100


2-] Regulator full opened: Speed MAX; Pressure=0 Pa
From Bernoulli: P + 1/2 density V²= K.(K is now known)
V= SQRT((K- P)/(1/2 density))
V= SQRT((98100-0)/(1/2 1000))= 14 m/s MAX FLOW SPEED.

3.] Any regulator possition.PRESSURE from 98100 to 0 (K is known)
V= SQRT((98100- P)/(1/2 1000))


In real situations, ideal conditions do not exist. The shape and texture of the tank, the valve, the pipe, their fittings, ...... everything leads to turbulence and drag, that will reduce the flow rate.

But, in anycase, the Bernoulli equation is used in one or another way.

This equation is a particular expresion of the Energy conservation principle relating the POTENTAL ENERGY (Pressure) with the KINETIK ENERGY(Flow speed).

Is this equation used in the case when a body is sorrounded by fluid ? Of course YES.

Every time a fluid has to move around a body, it has to accelerate.
Think on water around a ship. Water at the hull's sides moves faster than water at the bow becouse it has to sorround the ship.

If the fluid moves faster, then, as Bernoulli stated, pressure drops. Hence, there is a low pressure area that sorrounds a moving ship. The faster she moves, the lower that pressure.

That's why anything, too close to the hull of a fast ship, is attracted to her. The "thing" is falling into a low pressure "hole".


As told above, water close to the hull moves faster than water far from it.

But .... what are the speeds ?
How far does the hull influence extend through the water ?

A complete hydrodynamics problem is solved, when all flow velocities and pressures around the studied body are known.

The volume where fluid behaviour is influenced by a body inside it, is called the "field", a "speed field" when speaking about speeds and a "pressure field" when pressure is under study.

In further writtings, the difference between near and far fields will be also explained.



The example of the tank and the pipe is named an INTERNAL flow condition, while the example of the ship, is called an EXTERNAL flow condition.

Of course for INTERNAL flows, the "field" is limited by the container. The problem arrives when its required to perform an external flow anlysis.
For this later case, the extension of the field is theoretically as big as the whole fluid body. In the case of the ship, it ould be the whole ocean.
In order to allow for practical solutions, the field is only extended to a distance, at which the body's influence can be neglected.
This distance, as it will be explained, is not always the same. Depending on what's going to be solved, the extension of the field must be changed.
For practical solutions procedures, this reduced field is named the "DOMAIN". The choice of a correct DOMAIN size and shape, has a key relevance in the results.


Regards


#24

Before going on with the fluids properties and their relations, just a little addon about the Benoulli relation applied to EXTERNAL flows.

Remember what Bernoulli equation means. If speed of a fluid increases, pressure decreases. And, that it works both ways.

By now, we dont mind the perssure values, but only how they change.

Have a look to the attached pic:

There is a kind of wing, with a section shape (profile), drawn in RED, that is submerged into a fluid (air, water ....) that moves from left to right (BLUE ARROWS).

The profile is set in such a way, that makes an angle with the incoming flow, marked as ATTK. This is the angle of attack.
This angle is measured from the free flow direction. to the profile CHORD. Marked by the straight line from X to Y.

Point X is named the LEADING EDGE. The first point of the profile that meets the flow. Point Y is named the TRAILING EDGE, the last point of the profile in contact with the fluid.

The straight distance, from X to Y, is the CHORD LENGTH.



Let's watch the stream lines around the profile. The closer to the wing, the more diverted their paths result.
Those stream lines very close to the profile, have curved shapes, while those very far, are straight lines (the free flow direction). And ....... there is a more or less smooth transition between both shapes.
Lets have a closer look to the streamline between points A and B.


Flow, in this line, has to run around the upper curved face of the profile. Hence, it has to accelerate and run faster than the free flow.

Lets do the same with the stream line between C and D.
Flow beween C and D, is also diverted, but not as much as the former one, and hence, fluid along this line, runs slower than that at AB.

The influence of the wing in the fluid, extends to some distance, ahead, astern, over and below the profile. And then, speeds are modified all around the wing. (speeds field)

Fluid over the wing, is a lot more diverted than fluid under it, and hence, in bulk, fluid over the wind, moves faster than fluid under it.

Recalling the Bernoulli statement ............ If speed goes faster, pressure decreases.

Then: At the slower, lower side (light green area), there will be more pressure than at the faster, upper side (light orange area).(pressure field).
The pressure diference, and its distribution along the chord, lead to the development of a force, marked as F, that PULLS UP the wing.

Please note a key feature of hydrodynamic and aerodynamic shapes:
As, I hope, it's put in clear, we don't mind on speeds or pressures values by now, but only on the differences between pressures and speeds above and below the "wing".

Profiles are designed in such a way, that the maximum pressure differences are located at the desired place. From this point to the trailing edge, speeds, and hence pressures above and below, start to equalize.

In an ideal shape, both must be the same just when the fluid leaves the wing.

At the same time, pressure distribution (the shape of the low and high pressure lobes in light green and orange), must result in such a way, that the DRAG force gets minimized while the Lift force is maximized.

MAXIMUM LIFT WITH MINIMUN DRAG.


Please note that changing the angle of attack, the pressure lobes shape would change as well.

As pressures are developed due to the possition and shape of the profile, it's curves, it's proportions and dimensions, must be duly designed "THIS IS THE KEY IN FLUIDS DYNAMICS".

The resulting total force, that was marked as F, can be broken into two components:

L, the lift force.
D, the drag force.

Notice that not only the strenght, but the orientation of F, depends on pressures around the wing. Depending on how much this force is tilted, the distribution between DRAG and LIFT changes.

The tilting angle of this force, depends on the shape of the pressure lobes, and hence, on the profile design and angle of attack as well.

The normal question usually is:
Find the best shape and angle of attack, for the problem conditions (Speed, thrust, torque, weight, available volume limitations ....... and some more that are not so obvious, but that will be explained in later posts ....)



Kind regards:

NOTE: Bernoulli equations do not completly explain the question. In fact, do not fit with experimental results, and do not answer many questions. Anyway, as the basic principles (speed-pressure relation) are the same, it's a very good approach for understanding.


#25

Hi Mark:

Back home.

I've made some calculations and simulations looking for the best setting for the duct supports:

From my results .......

Three suppports at 120º, ahead of the propeller.

Propeller:

Graupner 2308/65 (Supposed as belonging to the Gawn standard series)

Diam: 65mm, Pitch 34mm

Speed of advance: 1.08 m/s

RPM: 1500

Water density: 999.7 Kg/m³

Propeller ,without supports nor ducts, thrust and torque, taken as reference 100%

Supports section.


Data table:

Gray row: Supports setting angle.

UNITS: Percentage of "free" propeller.


Conclusion:

Thrust is reduced by the supports for all setting angles.

If the supports are required for holding the duct in place .......

Best setting angle, for the proposed profile: a= -5º

Now checking with dome type profiles which are easier to build.

Regards


#26

No improvement found by changing to dome type profiles.


#27

Ion,
I do have the propellers will need to adjust housing internal diameter for best fit.
Currently in verification of assembly and debug may get to local pond for test run this week. After testing will build version 2 with revised lead in and shape of duct and supports. The O ring seal agenst the side of tube is working well no leaks.
Thanks again for all the information.

Mark




#28

Hi MarK:

For ducted propellers, the best site I've found is this one:

http://www.prop-shop.co.uk/products.php?cat=kort-nozzle-propellers#

They also have quite a big and "real" propellers catalogue, from "surface" props to highly skewed scimitar submarine props.

Their propellers really belong to the standarized series, and hence, the whole information can be found.

I got in contact with them. Seem to be a lot more serious and "technical" than the guys from Graupner.

Being the ROV a low speed, "towing" device, an accelerating duct, together with a big area ratio, low pitch propeller, seems to be the best choice.

Regards

NOTE: Dont hersitate asking for any technical information. I'll try to help as far as I can.


#29

Well:

How is a fluid defined ? What variables must be taken into account ?

The whole thing goes around the speed and pressure fields that are developed around the submerged body.

Pressures and speeds are related, but some more things have to be taken into account.

At the beggining, we spoke about viscosity, the internal force that prevents fluid "layers" to move with respect to the others.

The bigger the layers speed difference, the stronger the force exterted by viscosity.

The very first fluid area, in contact with the body, is stucked to the body's surface, being almost at rest.
Hence speed distribution changes around the body, due to the shape and to the distance from the "hull".

At the same time, density has a key role, as it measures the mass of the fluid volume unit, and hence it's inertia.

But density changes with temperature.

For gasses, or generally speaking, compresible fluids, density also changes with pressure, that at the same time is also linked with temperature.

The same happens with viscosity. It also chages with temperature and pressure.

In a fluid, pressure and temperature may be (in fact they are) different at each location, and hence no data can be taken as fixed.

We have a set of variables, speed, pressure, temperature, specific heat, thermal conductivity, viscosity, density, ............ all linked, all inter-connected, and all changing along time and location.

- Speed - Displacement of a point of the fluid per time unit.
- Pressure - Force exerted on a fluid point(surface unit) from any direction.
- Specific heat - Required energy for rising the mass unit of fluid temperature by one degree (Kelvin)
- Thermal conductivity - Bodies (includes fuids) capacity for allowing heat to be transmited through it.
- Density - Mass per volume unit.
- Viscosity - Resistance of fluids to pour.

................ there are more ...... ;-)



At the same time, a change on any of them, in one fluid portion, will modify all of them, not only in the portion, but in the whole fluid around it. Making things even more complex, those changes require a time that, at the same time, depends on the fluid conditions.

In short words, a real unsolvable mess.


Fortunatelly, for practical applications, some of those variables can be neglected or supposed to be constant, allowing for the solution of the problem.


What to neglect ? What to take as constant ? The answer will depend on each problem and what is wanted to be solved.

First thing to think on, is .... Are we dealing with a compressible or with an uncompessible fluid ?
What is the same. For the fluid under study, Does pressure have a relevant effect on the fluid density ?
If the fluid is air, the answer is YES. But if the fluid is water, the answer, for most applications is NOT.

Viscosity is another thing that we could neglect under some conditions.

Two things: Viscosity can be taken as constant or completly neglected. It depends .........
While a fluid moves, due to its mass, inertia is playing a role, that "wants" to keep things as they are. Inertia works against any change in the motion (direction and speed).
At the same time, viscosity is trying to stop the fluid. As it depends on speed differences (prevents fluid layers from sliding one on the others), it mostly works closed to the hull.
If INERTIA is a lot stronger than the viscosity effects, viscosity can be neglected, if not, it must be taken into account.
That "A lot stronger", is not a fixed quatity, it will depend on what is going to be calculated and on the enviromental conditions.

Properties changes with temperature, can be neglected if under the problem conditions, are very little.
Once again, the word "little" means, that for our problem, and desired answers, those changes have no relevance.

Hence, all variables are connected, so, a conceptual simplification of the problem is the first thing to be done.
The real danger when dealing with fluids dynamics is, that even a wrong mental model(simplification) will return results, that with no knowledge, could be taken as good.


Regards.



#30

Hi Mark:

Some datas for your project:

Test speed 2.2 knots.

Both propellers working at 1500 RPM

Pressure coefficent distribution:

Please note that both curves are slightly different. That's due to the battery tubes supports. The difference will develop a vertical force (downwards in this case).

By integration of both curves, the Dynamic Center of Pressure can be found (DCp).

By checking the relative possitions of DCp, CG(center of gravity) and CB(Center of buoyancy), the behaviour of your model could be duly studied.

Note that DCP moves when speed changes.

Rt(Total resistance)= 5.028N

Vertical force= -0.6N

Regards



#31

Hi,Ion
@Ion , I would like to design an autopilot controller for my openrov (using a state-space model), but I can’t find the coefficients needed to get started (friction matrix/drag coef, added mass, coriolis forces…).
Do you have it , do they come from simulations or tank-testing? Can you send the hydrodynamic coefficient of Openrov 2.8 to me.
Thanks a lot for your help!


#32

Ion

I just today saw this thread today and am reading it for the first time. As I go through it, I am creating notes / questions / comments so please bear with me.

Post #4 This level of technicality I was able to understand and follow. (I hold a BA in Applied Science)

Post #9 Your technical level and use of unfamiliar terms is starting to lose me. If I understand your comments and illustration correctly, a cone shaped leading edge is preferred over a round leading edge which in itself is preferable over a flat shape? This is more critical in the following edge than the leading edge to reduce drag?

Post #14 If I am reading this correctly, a dimpled surface would produce less drag than a smooth surface? This seems contrary to conventional thinking and practices. Racing boat hulls are waxed and polished to minimize drag on the water. To what extreme should the dimple be? Suppose I use a 1/16" diameter drill bit with a 1/16" rounded nose drilled to 1/16" into the surface, compared to a 1/2" with a 1/2" rounded surface drilled 1/2" deep into the surface? Is it a question of speed that would influence the answer? Should these be in a straight square pattern (such as the dots on a 9 on a dice) or should the dimples be offset pattern? (such as the dots on a 5 dice)

Post #19 Back to a technical level that I comprehend.

Post #30 Thank you for taking the time to write this tutorial. I found it very helpful in better understanding the topic. One more question remains in my mind. Given the increased pressure with increased depth, theoretically, drag on the object moving through the water would increase with the depth. This is probably a miniscule impact on the outcome but in theory would be measurable?


#33

Hi all:
WANGJIA>> I have those coefficients, or … I think I have them. Its so long since I made those calculations, that I must look for them in my backup unit. Give me a few days.
As per your second questions. The numbers come from a couple of hand and CFD calculations. No tank testings involved.

JIM_SHOLTZ >> Its hard to find the posts you are asking for. May you quote a few starting words from each one in order for me to be able to find them ?
Anyway … Best bow shape for underwater vehicles is elliptical. If the hull or apendexes are going to be wing shaped(from a profile extrusion), the best leading edge shape, is closer to an ellipse than to any other form. Anyway, no hydrodynamic profile leading edge is in fact elliptical, but very similar.
Yes, the trailing edge shape is critical. It must be designed in such a way, that all arriving flows will have parallel vectors and the same intesity, or what is the same, all velocity vectors at the trailing edge must be equal. When the body starts moving, a series of eddies are induced, (Van Helmholtz inestabilty), due to the irregular velocity distribution, but as soon as the stationary regime is achieved, the outflow at the trailing edge would have to become steady and parallel.
Well, the surface texture question is really interesting. The goal is preventing the boundary layer from being stucked to the hull, and hence minimizing the viscous drag. This is achieved by granting a condition where the boundary layer is turbulent while sorrouded by a pseudo laminar flow. I say “pseudo”, becouse a Laminar condition is not possible to be achieved into those conditions.
Remember a shark skin or those full body swiming suits that were forbidden in high level competitions. The way to go is not drilling on the surface, but smoothly sanding it.
At my sailing club, and I think that even in most Class rules, it is forbidden to give such “sanded” surface to our boats(ripplet effect).

Water is an almost perfect Newtonian fluid so, density and viscosity do not depend on pressure. Therefore, drag is not dependant on depth. The only thing that could be noted, is related to cavitation concerns.


#34

I’m very glad to know you have these values,if you find these values,I hope you can let me know soon.Thanks a lot for your help.


#35

Im looking for them. Anyway, Are you trying to build a mathematical model of the ROV dynamics ? Or just an Autopilot ?


#36

HI Ion,
Thanks for your reply,I want to build the dynamic model of the ROV,so I need the hydrodynamic coefficient(friction matrix/drag coef, added mass, coriolis forces… ) ,thanks for your help,best regards to you.
/J wong

Ion
November 2 |

Im looking for them. Anyway, Are you trying to build a mathematical model of the ROV dynamics ? Or just an Autopilot ?

Visit Topic or reply to this email to respond.

In Reply To

wangjia15623574012
November 2 |
I’m very glad to know you have these values,if you find these values,I hope you can let me know soon.Thanks a lot for your help.

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#37

For a dynamic model, you’ll need the ROV’s RAOs. Or will you try to compute them from scratch ? Quite a hard work ¡


#38

Check this post of mine from 2014. Forces can be turned into Coefficients just using any adimensionalization parameter.

Most comon is: Cd= 2 x Force/(density x Section area (Or any reference surface area) x speed²)

Regards


#39

Hi,Ion
Thanks for your reply。I have read it ,But I don’t know how I can get the hydrodynamic coefficient(friction matrix/drag coef, added mass, coriolis forces…) from the force in Y axis,can you explain it in detail?Thanks a lot for your help!
/J Wong


#40

Dear Wangjia:
I like teaching, but I think making a little bit research will be better for you to improve your knowledge. Obtaining coefficients from forces is just as basic for hydrodynamics as adding is for general maths.

Studying and researching is the only way to achieve knowledge.

Regards