Chapter 2

The Mathematical Models of Fluid Motion

2.1 Preliminary Remarks

Analyzing the similarity conditions in a commonly used model of an incompressible viscous fluid and the corresponding system of Navier-Stokes equations, two fundamental questions are raised.

1. In conditions of turbulent flow it becomes necessary to admit the nonuniqueness of solutions of the Navier-Stokes deterministic initial data and corresponding random results. For example, when serving in the tray of some form of a fixed water flow in the turbulent flow regime in the fixed point of the observed changes (fluctuations) of the velocity and pressure, slimming information and considering the average (by chance) values of the velocity and pressure, lose the possibility of practical use of the Navier-Stokes equations, as their averaged similar Reynolds equations are open. The system of criteria of similarity derived from open-loop equations, obviously, may be incomplete. But which way is advisable to seek ways circuit equations: reducing the scale of consideration and increasing the detail of the description by clicking on the molecular level with the equations of Boltzmann; or on the contrary, enlarging the scope of the consideration by the spatial averaging effects and reducing the number of measurements essential to the problem?

2. Flows in real borders are not clear to the question of boundary conditions for the Navier-Stokes equations with turbulent flow. What is the boundary of the flow in the channel with naturally uneven surface or in the river, where the bed is composed of permeable eroded material? Even in a glass tube with a perfectly flat wall at the turbulent regime is there a need to prevent "sticking" to the wall or slipping fluids?

These issues are significant for detailed studies of turbulent flows. A large number of practical problems can be solved with the help of physical experiment, based on more rough mathematical models of fluid flow.

Consider the following models of real unsteady flow:

In addition to these models, we will consider the equation of turbulent flows, averaged on specially selected volumes (two-dimensional, one-dimensional zero-dimensional equations). We show that the degree of open-ended equations is significantly reduced.

The first model of the flow may be implemented only under the condition of a slip fluid impermeable border. In contrast, all other models allow for the formulation of the problem with both slip and the condition of "sticking" of the fluid to a rigid border.

Modeling of riverbed flows also needs to take into account the deformation of the riverbed by the migration of soil particles. In this regard, the system of equations of fluid should be supplemented by the terms of the start (and stop) strains of the riverbed in sediment transport and the equations that define the process of erosion, weighing and deposition of particles. Here also it is possible (as appropriate) to increase difficulty of models from the model kvazi one-dimensional environment (dispersed), in which the presence of the sediment is taken into account only after the conditional density single-phase environment (water + the soil particles), and ending with complex models of the dynamics of a two-phase medium in which turbulent motion of each phase is described separately, taking into account the interaction of phases.

2.2 Conditions of Mass and Momentum Conservation

The condition of conservation of mass for a multiphase environment can be written in the integral form

(2.1)

where the integral is taken by volume Wi changing the shape (and dimensions), but consists of the same particles of the i-th environment, having a density of ?i. This condition can be written in the differential divergence form

(2.2)

where Ui is the true velocity of the considered phases.

Equations of mass conservation are almost the same for all models of motion. The equations of conservation of momentum, written in stresses for a fixed volume (or mass) and expressing Newton's second law, have the same form for all models of motion too:

(2.3)

(2.4)

Here Wi as in (2.4), the volume occupied by the same particle of i-th environment; Oi is the surface limiting this volume; Fi is the force per unit mass; sn is the stress vector on the site dO with the normal n; sn={skj}n, where {skj} is the stress tensor.

Usually it is not easy to monitor the volume occupied by the same particle. It is easier to lock in a fixed amount Wio, limited surface Oi0. In this case, the equation of conservation of momentum becomes:

(2.5)

where n is the external normal to the surface Oi0

Differential analog of equation (2.3) in divergence form does not exist. To move from the surface of the integral in the right-hand side of (2.3) to volume is only differentiability of stress fields sn:

(2.6)

(2.7)

The stress tensor of a fluid can be written as a sum of a spherical tensor {s} and deviator {s'kj}

(2.8)

where is the first invariant tensor voltage (average normal stress); compression s<O.

The value ?=-s is usually called pressure. Pay attention to the fact that this pressure is not necessarily the same as the pressure P defined in thermodynamics and characterizing the state of the environment.

In a scalar form, equation (2.6) is:

(2.9)

On repeated indices the summation is executed.

The difference in mathematical models begins to expand links between the stress tensor of a fluid {skj} and kinematic characteristics of the fluid - velocity fields U(x).

The energy conservation law when considering only one kind of energy - mechanical - does not introduce any additional information, as it is obtained directly from the second law of Newton. This is not the case when taking into account all forms of energy, in particular the internal energy of the fluid. In this case, the equation of energy balance (in all forms) is necessary.

Applying the similarity transformation to (2.2), make sure that it leads to the generalized criterion of Strouhal:

(2.10)

and criteria that include the extent of the density,

(2.11)

Here t0, U0t, and l0j are typical values of time, velocity, and length (in j direction); ?0, ??0 signifies the characteristic values of density of the environment and its changes. Upon receipt of (2.10), (2.11) is accepted that the magnitude of velocity change must be equal to the scale of velocity. On the contrary, for a difference of density and density values it is not the requirement, though possible.

The same transformation of equation (2.9) leads to the following criteria:

(2.12)

(2.13)

(2.14)

(2.15)

(2.16)

(2.17)

Equation (2.14) is the criterion of external forces; in the gravity field becomes the Froude criterion; (2.15) - the Euler criterion; the relation (2.16) - criterion internal deviatoric stresses for viscous fluid transformed into the Reynolds criterion. The symbol ?? emphasizes that if s=-?, then the pressure in (2.9) is determined only up to a constant.

2.3 Non-Viscous Fluid

Model residual fluid, usually unsuccessfully called "ideal," gives excellent results in the conditions when the energy dissipation (transition mechanical energy into heat) is insignificant, and the main phenomena are related to the inertial forces and the forces of pressure. Inadequcy of the term "perfect" that any mathematical model of fluid in a sense, is ideal.

In residual fluid vector stresses sn collinear to the vector n, normal to the surface O in areas where the vector exists, i.e. outside of singular points. This means that at any place, at any point, except for the singular, the instantaneous value of the tangent stress is always zero, and the value of the normal stress does not depend on the orientation of the area. The stress tensor has the ball form.

Believing all off-diagonal components of the tensor {skj} are equal to zero, we obtain instead of (2.6) and (2.9):

(2.18)

If you enter the rotor (vortex) velocity rot U=?×U, the left part of (2.18) can be converted emitting terms that explicitly depend on vorticity...