## BASICS OF THE THEORY OF HYDRODYNAMIC SIMILARITY. MATHEMATICAL MODELING

After studying chapter 5, the bachelor must:

* know *

• basic concepts and definitions of the similarity theory;

• the physical meaning of the similarity criteria;

• dimensional analysis method;

• the basics of mathematical modeling;

• mathematical statements of boundary problems of hydromechanics;

• the difference between parabolic equations and hyperbolic equations;

* be able to *

• use the theory of similarity in solving practical problems;

• apply theorems of similarity theory when performing mathematical modeling;

• find solutions to boundary value problems using the method of separation of variables and integral methods;

* own *

• mathematical apparatus for solving boundary value problems of hydromechanics.

There are two methods for studying physical phenomena - analytical and experimental. In the analytic study of fluid motion, the problem reduces to integrating a system of differential equations under given conditions of single-valuedness. For example, for a viscous incompressible fluid, we have a system of differential equations

(5.1)

(5.2)

where (5.1) is the system of Navier-Stokes equations written in vector form (see paragraph 4.9); (5.2) is the continuity equation. In addition, the initial and boundary conditions and the values of the physical constants ρ and * v * must be specified.

In principle, the aggregate system of basic differential equations and the conditions for single-valuedness, the concrete single phenomenon is completely defined. However, these equations are extremely complicated (they are partial differential equations), and solutions have been found only for a small number of particular cases, and for very important simplifying assumptions.

Another method of investigation is a direct experiment. In this case, those quantities that are of direct practical interest are measured, and connections that allow direct application are found. However, the data obtained from the experiment will relate only to the particular case for which the experiment was performed. It is necessary to find ways of generalizing the data of experience on other related phenomena. This would allow us to judge on the basis of a few experiments the fluid parameters in numerous related phenomena. The task of finding a scientifically grounded method of generalizing the experimental data is solved by a similarity theory, which is the theory of methods of generalizing the experimental data.

## Basic concepts and definitions of the similarity theory

We give some definitions of the similarity theory.

By * a class of phenomena * is meant a system of differential equations describing a physical phenomenon. For example, the system of Navier-Stokes equations and the continuity equation describe all possible types of motion of a viscous incompressible fluid in channels of any shape.

By * single phenomenon * is meant a system of differential equations with uniqueness conditions imposed on it.

By * a group of phenomena * is meant a system of differential equations with similar uniqueness conditions imposed on it. For example, the phenomena occurring in channels, geometrically similar, will belong to one group of phenomena.

The basic idea of the theory of similarity is to distinguish within a class of phenomena of narrower groups.

* Similar phenomena * are those in which the ratio of the variables characterizing them is a constant number. There are the following types of similarity.

1. In order for the model to be mechanically similar to the sample (the object for which the model is created), first of all, * geometric similarity *; for this, the ratio of the lengths of the similar segments of the sample and the model should be the same, i.e.

where - some linear size of the model flow; - the corresponding flow size in the sample; - the constant of geometric similarity (linear scale of the model).

The last formula also implies the relations

where - the area of the model and the sample, - the volume of the model and the sample, respectively.

2. In obtaining the model, in addition to geometric similarity, it is necessary to observe another * dynamic similarity *, which means that all the forces that cause the considered motions in the model must be changed with similar forces in the sample into one and the same number of times.

The force * F * is defined as the product of the mass

*for the acceleration*

**t**

**a****strong> , i.e. Since the dimension of the mass , and the acceleration , the dimensionality of the force will be <**

** It follows that for dynamic similarity, the ratio **

**(5.3)**

where ; ; ; - constant of dynamic similarity (scale of forces).

Condition (5.3) is a mathematical expression of the general law of dynamic similarity, which was first formulated by I. Newton.

In the theory of similarity, it is proved that when performing geometric and dynamic similarity, the kinematic similarity will also be observed.

In the case when friction forces prevail from the acting forces, then from Newton's law for tangent stress one can get that . Hence, taking into account the formula (5.3) and the relation , we get

The latter relationship represents the * Reynolds Dynamic Reaction Condition * under the action of internal friction forces.

If the influence of viscosity (frictional forces) is insignificant and fluid motion occurs mainly under the influence of gravity, then , where * g * is the acceleration due to gravity.

Relation (5.3) in this case takes the form

The last relationship is called * the law of Froude's likeness. *

Thus, for two similar phenomena, there must exist relations of the type

etc,

where keep constant values at corresponding points of similar systems. Therefore, these quantities are called * similarity constants. *

Generally speaking, there are not two such phenomena, but an infinitely large number. These phenomena constitute a group of similar phenomena. Therefore, an expression of the form is a group transformation of phenomena, where takes successively constant values when passing from one phenomenon to another like the first.

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