The liquids and gases have many common physical properties, because of that they are called *fluids*. Molecules of fluids have no fixed places in volume and they can move each other when applied small external forces. The fluids are losing shape comparatively easy. The study of fluids is made easier if is neglected their viscosity (their internal friction) i.e. if it is presumed that they are the *ideal*. The mathematical theory of ideal fluids allows to be theoretical explain a set of phenomena, connected with waves, lifting forces and induced drag during motion of a wing in the fluid. This theory can not explain many others phenomena and effects such as drag due friction, detached of flow from the surface of the objects, heating up of the fluids, heat and mass transfer processes, loses of a pressure when fluid is moves in tubs.
The modern applications of hydrodynamics and aerodynamics impose considering more complicated mechanical and mathematical models and more complete admit the real properties of fluids. At the same time are developed the mathematical models, where are admitted liquid to liquid or gas to liquid share surfaces. The absence of a symmetry of cohesion forces that act from two sides of the boundary lead to arise a *surface tension*.
The base part in fluid mechanics takes place the stress tensor and the rate-of-strain tensor. The connection between them can be different. The fluids for which the connection between stress tensor and rate-of-strain tensor is linear are called *Newtonian fluids*. Lots of liquids – water, glycerin, liquid metals, as well as all gases are Newtonian fluids. The fluids for which the connection between stress tensor and rate-of-strain tensor is non linear are called non Newtonian. For example such fluids are solutions of polymers, non-drip paints, tomato ketchup, blood and many others.
In hydrodynamics and aerodynamics are essential studding of Newtonian fluids and we are focused on studding only Newtonian fluids.
The base method of studding fluid flow is constructing of phenomenological macroscopic theories based on common hypotheses got from experience. Originating of phenomenological theory for fluid flow is based on the base three low of mechanics – lows of conservation of mass, momentum and energy. Let us assume that these lows are applicable for fluid flow and using some hypotheses that are specific for different fluids we obtain a closed system (a model) from partial differential equations (PDE). It is necessary to put the specific boundary conditions and so got boundary problems must be solved analytical or numerical. The full investigation includes the experimental proof of the results. While physical experimentation gives more accurate, realistic results which include nonlinear effects, it is often laborious, costly and time-consuming, and this is where computational fluid dynamics (CFD) and numerical modeling and testing enter the contribution. While computer will likely never be fast enough to meet everyone’s desires, today’s numerical processing capability lends itself to more complex solvers and accordingly more accurate solutions, with lower cost and eventually less labor and time.
CFD stands for Computational Fluid Dynamics. It is a complex of methods of computer modeling that, if applied well, can recreate the real-world behaviors of liquids and gases in a virtual environment. When CFD is applied to wind engineering, it is called computational wind engineering, or CWE. Fluid dynamics is a branch of mechanics concerned with the motion of a fluid continuum under the action of applied forces. Various mathematical models are used to describe fluid flow under different restrictive assumptions.
One of the hypotheses in fluid flow is that the fluids are not a discrete system of material points but they are continuously medium, i.e. they are a material continuum in which on every point is compared respectively a number that describe the density of the fluids.
To obtain of the Navier –Stokes equations that described the flow of viscose fluid it is used constitutive equations giving relations between stresses tensor:
and rate-of-strain tensor:
which can present as follow:
In this tensor equation are the unity tensor, and and are scalars. The coefficient is called *dynamic viscosity* of a fluid and the scalar is called *pressure*. We note that the tensors and are symmetric.
If , , are the unity vectors of the Cartesian coordinate system * **x, y, z *the stress tensor can be present as:
From definition of an active pressure on surface with normal it follows that the pressure is connected with two directions – direction of pressure vector and direction of unity vector . Components of the rate-of-strain tensor are connected with the components of velocity , , , as follows:
,
,
.
In this way the tensor equation can be written using follow equations:
*(i, j = *1, 2, 3*)*
where is the Kronecker symbol, i.e.
Using these constitutive equations it is obtained Navier –Stokes equation, which describes the flow of incompressible viscose fluids with density , kinematics viscosity , pressure p and velocity :
Where is mass force, acting on the fluid that is frequently the gravity force = . This equation expresses the law of momentum conservation. The law of mass conservation is defined by continuity equation that for incompressible fluids can be written as follow:
When it is considered a two dimensional problem, i.e. independent variables are two can be introduced stream function thus to satisfies the two dimensional continuity equation. For example in Cartesian coordinate system the continuity equation for two dimensional problem is written as follow:
This equation satisfies identity if we put:
, ,
and vorticity for 2D case definition is:
,
where: *u* and *v* are velocity components along *x* and *y* coordinates. We can combine these definitions with Navier-Stoke equation. It will eliminate pressure from Navier-Stoke momentum equation. That combination will give us non-pressure vorticity transport equation which in non-steady form can be written as follows:
,
where: is the Reynolds number and *a* is a characteristic length.
**Boundary conditions for viscose flow.** The system from partial differential equations consists of Navier-Stoke momentum and continuity equations have any number of partial solutions. If it is known one of them we can try to obtain theoretically the flow that corresponds to it. When it is used this method it is possible to obtain useful information, but not always for given partial solution correspond a real fluid flow. The base method of approach in fluid mechanics is to obtain those solutions which correspond to considered flow. For this purpose it is necessarily to assign initial and boundary conditions, which are satisfied from requested solution.These conditions give additional information about character of mechanical processes on the boundary *S* of a domain *G*_{1} taken from a fluid, and also initial condition of a fluid.
When a body moves in a fluid (liquid or gas ) arise forces - drag and lift. Drag and lift are aerodynamics forces that are generated by the interaction and contact of the solid body with the fluid. Computing the forces on a body moving through a viscous fluid are the main goal of computational fluid dynamics. These forces cause stresses in the body. If the stresses are sufficient the body become deformed. |