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Viscous Fluids & Potential Flow · Acedra
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Viscous Fluids & Potential Flow
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0:02
Hello welcome to my talk, All about Fluids. this talk is on an interesting topic of whether viscose a fluid could form a potential flow? this is the first part of the talk with the focus on the theoretical aspect, relevant to the topic. As you all know it very well for potential flow, the fluid is required to be irrotational and inviscid. and in many cases the flow is also assumed to be incompressible the question is, 'do we need all these assumptions for
0:58
an irrotational flow?' I will answer the question in the talk let's start from the full Navier-Stokes equation we can see in the full Navier-Stokes equation, in the left hand side, the substantial derivative is used for calculating the flow acceleration which consists of two parts: the local acceleration and the convective acceleration in the right hand side, the force due to the flow pressure; the body force acting on the fluid; the dissipation term due to the fluid viscosity and the fluid
1:52
compressibility term. the last two terms are related to the fluid viscosity. for an inviscid fluid, the fluid viscosity coefficient MU equals to zero, thus the Navier-Stokes equation is degenerated to Euler equation although Euler equation is not perfect, it is very useful for deriving the Bernoulli's equation, which is actually an integral form of Euler equation under the assumptions of an inviscid and steady flow this is a conventional assumption for the inviscid flow
2:48
in this slide, I would like to clarify the concepts: an inviscid fluid and inviscid flow and this clarification of these two concepts is very important for irrotational flows an inviscid fluid is a fluid with zero viscosity, that is, MU=0 This is an idea assumption since all fluids are viscous, however for many practical applications, this assumption is acceptable because it could simplify the fluid dynamics a great deal, while it still gives a very reliable and good
3:49
approximations to the problems. An inviscid flow is a flow for which the last two terms in the full Navier-Stokes equation vanish Obviously, an inviscid fluid will cause an inviscid flow, however a viscous fluid could also cause an inviscid flow, such as the uniform flows and the incompressible irrotational flows, you will see this clearly later in the talk
4:33
Conventionally, there are two basic hypotheses for a potential flow as the following hypotheses: 1 and 2: the flow is an irrotational flow, this assumption would lead to a velocity potential function, and under such an assumption, the three velocity components are no longer independently each other. the mathematical expression for the irrotational flow is the curl of the velocity vector is 0 or its circulation of a closed curve is 0, the second assumption is inviscid fluid, MU=0, and
5:25
the corresponding fluid dynamics equation is Euler equation. however in many practical problems, the flow is also assumed as incompressible this assumption could simplify the problem so for finding the solution easier. This assumption is made for the purpose of simplifying the problem rather than a pre-requisite, because we have seen in many cases the compressible flows can be potential flows too, so the question is: for a potential flow, do we need all these assumptions?
6:18
in this slide I will show you what is the potential flow. in principle potential flow is a irrotational flow, because this assumption could lead to the velocity potential function and there should be no more assumptions needed for a potential flow, then what is the irrotational flow? the definition of a potential flow, the circulation of fluid velocity vector on an arbitrary closed curve, is 0, for instance, for the arbitrary closed curve in the fluid, C,
7:01
here its circulation is calculated as this. if the circulation is zero, then the flow is irrotational. Based on the Stokes theorem, we can convert the integral on a closed curve to a surface of the enclosed curve and from the zero circulation, we can obtain another definition of irrotational flow, as this, the curl of the velocity vector is zero
7:42
the physical definition of an irrotational flow is that the fluid elements do not rotate around their own axes, while moving along the pathlines as this. this is similar to the famous Ferris wheel, on which the capsules are hinged to the large rotating wheel. when the wheel rotates, the capsules keep their upright position all the time, and the passengers in the capsules could not feel any rotation. the velocities of the different positions on the capsule are
8:30
all same, same as the velocity at the hinge. for a rotational flow, the fluid elements rotate around their own axes, while moving along the pathline as this
8:52
in reality, how we can tell which flow is the rotational or irrotational flow. Generally, there are two ways to check whether the flow is irrotational or not the first one for checking whether the curl of the flow velocity vector is zero the second way is to make a closed curve, any closed curve in the fluid if you like, and check whether the circulation is a zero. here take the Couette flow as an example its velocity distribution is a given as this
9:47
and the curl of the velocity vector given by this, it is not zero and if we draw rectangle in the Couette flow and check whether its circulation is zero. Obviously in this Couette flow, its circulation is not zero, thus we can see both methods have shown the Couette flow is rotational. now we may have a question, where we can see the rotational and irrotational flows in practical flows? here a uniform flow passes a plate and it can be seen near the plate surface, there will be a boundary
10:51
layer where the flow velocity increases from zero at the plate surface due to the no-slip condition to the same velocity as that of the uniform flow, this layer is called boundary layer. if we extend the flow to a large range of the flow as this, so we can see in the boundary layer, we draw a rectangle and we can see its circulation is not zero, thus the flow within the boundary layer is rotational. and outside of the boundary layer, we can see the circulation on a rectangle is zero
11:48
thus the flow outside the boundary layer is irrotational. Here we look at the famous example, the flow in a horizontal pipe as this so when the Reynolds number is small, such as, less than 2300, and the flow in the pipe is laminar, and we can obtain the analytical solution for the fluid velocity distribution as this. This flow is also called Hagen-Poiseuille flow, and it can be seen such a flow is a rotational, however with the
12:40
fluid Reynolds number is increased, say to ten million, and the flow in the pipe is fully turbulent and the flow velocity profile is something like this, as such the large portion of the flow is irrotational, only the regions near the pipe walls here and here are the viscous boundary layers and flow within the boundary layer are rotational. When the Reynolds number become very large and the boundary layer would become thinner, as such the entire
13:31
flow might be taken as an irrotational flow for a good approximation, this is a good practice in many practical problems
13:48
for an irrotational flow, the zero curl of the velocity vector would lead to a velocity potential function, Phi and the fluid velocity vector of three independent components are transformed to a scalar function, Phi, and this assumption of irrotational flow could simplify the dynamic problem very much in a Cartesian coordinate, the three velocity components are calculated from the velocity potential function as this, so if we obtain the velocity potential
14:39
function and then we can calculate the velocity components, use these three equations
14:53
if the flow is incompressible, the corresponding continuity equation could lead to Laplace equation for the velocity potential function. we have seen many applications by solving the Laplace equation in the practical problems This is generally acceptable when the flow velocity is small, for instance, less than 0.3 Mach number if the flow is compressible, the continuity equation would lead to a partial differential equation for the velocity potential function as this
15:45
so we can see this partial differential equation for the velocity potential function is much more complicated than the Laplace equation. and also here we have one more unknown, the fluid density, and we also need to solve the fluid density in the dynamics. thus we need a one more equation for the problem, for instance, we may use the idea gas equation if the compressible flow is steady and the assumptions of an idea gas, and small disturbance in the flow field as this, so we can get the partial
16:46
differential equation for the velocity potential function, Phi, as this. Here M is the Mach number. For more details how we can get this partial differential equation for the velocity potential function can be found in the book, 'Modern Compressible Flow' of John D Anderson.
17:16
from the full Navier-Stokes equation, we can see, to obtain an inviscid flow, the last two terms must vanish. For this purpose the simple analyses are made for these two terms. to make sure these equations are satisfied the velocity vector gradient tensor or the velocity vector divergence are 0, for instance, a uniform flow can satisfy this. To ensure the term of this and this are both zero, this can be satisfied in incompressible irrotational flow, since for
18:09
incompressible irrotational flows, we have this and this, however, for compressible flows, these two terms are not zero. Now we look at the last two terms in the full Navier-Stokes equation for irrotational flows, the last two terms can be written as this, and put these together, we have equations as this. and for the convective acceleration we can use the vector calculus identity, as this. Consider the flow is an irrotational
18:57
flow, so we can have this, if the fluid gravitational force is the only body force, we can put all this together, we have the equation for the irrotational flows as this
19:17
so the integral form for the irrotational flow is given by this this is the general Bernoulli's equation, under the only assumption of irrotational flow, here we don't make the assumptions for inviscid flow or incompressible flow. if the irrotational flow is incompressible, then the term related to the fluid the viscosity disappears so we have the conventional Bernoulli's equation for irrotational flow this is the same equation as we seen in many textbooks, however for deriving
20:07
this Bernoulli's equation, the assumption of an inviscid flow is not required. and the Bernoulli's equation is automatically satisfied when the irrotational flow is assumed to be incompressible
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