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48 3 Basic Equations

In order to describe a complete flow field, this approach is applied to all points x& in space. Such a description is called a Eulerian description.

In differential equations, often the time derivate is used. Using the Lagrangian description, the time derivate of F is

This so-called substantial or material derivate describes the complete derivate of the property F of a liquid element. Remember that all flow quantities are only a function of particle and time. In contrast to this, the Eulerian approach describes the flow quantities F as function of point x& = (x1, x2, x3) in space and time t. Us-

In order to emphasize that the material derivate is meant, the expression DF/dt is usually used instead of dF/dt. Because dxi /dt = ui, the material derivate now reads

where u& = (u1, u2, u3) is the velocity vector. The Lagrangian and the Eulerian approaches, if applied to a special problem, must have the same overall result. For example, the resulting acceleration of a liquid element passing any point in space (Lagrangian description) must be identical to the resulting acceleration at this point predicted by the Eulerian approach. In order to explain the different terms in Eq. 3.6, the flow through a converging nozzle shall be regarded. Due to the reduction of cross-sectional area, the flow velocity is larger at an upstream position (point A) than at a downstream position (point B). The first partial derivate on the right hand side of Eq. 3.6 is the local derivate. It describes the temporal change of F at a fixed point in space and is equal to zero at points A and B if the flow is steady. However, the flow velocity at point B is always larger than at point A, and a liquid element passing point A will experience an additional acceleration which is not included in the local derivate. This additional acceleration is expressed by the remaining three terms on the right hand side of Eq. 3.6, the so-called convective derivate. Hence, the Eulerian description enables us to distinguish between a local and a convective part of the material derivate, while the Lagrangian description does not.

50 3 Basic Equations

passed by the flow without resistance must be found. Such a fixed control volume (cv), also shown in Fig. 3.1, equals the volume of the system at time t. Using the masses contained in the volumes 1, 2, and 3 at times t and t + t, the left hand side of Eq. 3.10 becomes

In order to find expressions for the masses m3 (t+ t) and m1 (t+ t) contained in the volumes 1 and 3, the volumes V1 and V3 are needed. These volumes can be calculated from

Fig. 3.1. System-to-control-volume transformation: system moving through a control volume, two-dimensional flow and control volume

Remember that the unit vector n& is normal to the surface element dA and always points out of the control volume (cv). Because A3 plus A1 completely surrounds the control volume (A3 + A1 = cv), Eq. 3.11 is equivalent to

The left hand side of this equation describes the rate of change of mass in a Lagrangian frame of reference, while the right hand side represents the respective Eulerian description for a fixed control volume. Hence, Eq. 3.14 is the desired sys- tem-to-control-volume transformation.

A more general expression for such a transformation is given by the Reynolds transport theorem,

where Nsys is the extensive quantity (e.g. mass, energy, momentum) and Κ is the intensive property associated with Nsys, the property per unit mass. The time derivate of the first term on the right hand side can be moved inside the integral, since for a fixed control volume the limits on the volume integral are independent of time.

Combining Eqs. 3.10 and 3.14 results in the continuity equation in integral form:

For steady flow, the left hand side is equal to zero. Assuming that the velocity is also normal to all surfaces where fluid crosses (e.g. pipe flow, Fig. 3.2) results in

where u&i ui .

If the densities and velocities are uniform over their respective areas, the flow can be treated as one-dimensional, and the continuity equation finally reads

Fig. 3.2. Conservation of mass: one-dimensional steady pipe flow

52 3 Basic Equations

3.1.2.2 Conservation of Momentum

Newton’s second law, the momentum equation, states that the sum of all external forces acting on a system equals the rate of change of momentum of the system:

forces acting on the surfaces of the control volume (pressure, shear, additional surface forces from a solid wall) and body forces (e.g. gravity) acting on the mass inside the control volume.

Eq. 3.20 can be simplified significantly if the control volume has entrances and exits across which the flow quantities may be assumed to be uniform, and if the

where N is the number of exit and entrance areas. If there is only one entrance and one exit, and if the velocity vector is normal to the entrance and exit areas, the momentum equation reduces to

Fig. 3.3. Momentum equation: forces acting on a control volume

At the entrance u& n& u , since n& and u& point in opposite directions. Note that the momentum equation is a vector equation and consists of one equation for each coordinate direction. If only external forces caused by static pressure and the wall are considered, Fig. 3.3, Eq. 3.22 reads

3.1.2.3 Conservation of Energy

Many problems involving fluid motion such as compressible and non-isothermal flows require the use of the energy equation. The energy equation for a system reads

where dQ/dt is the amount of heat per unit time and dW/dt is the work per unit time transferred to the system. Et is the total energy consisting of internal energy ΥeV (V: volume of the system, e: internal energy per unit mass), kinetic energy VΥu2/2, and potential energy -VΥ g& x& ( x& : position of system at time t, g& : vector

of gravitational acceleration). In terms of a control volume, Eq. 3.24 becomes

by applying Eq. 3.15, where Nsys = Et, Κ = et , and

The work-rate term includes the work on the boundaries due to pressure and tangential stresses plus the work added by source terms (e.g. shaft work) minus the reduction due to energy dissipation. The heat-rate term includes the rate-of-energy transfer across the control face due to a temperature difference and the energy production or reduction inside the control volume due to energy sources or sinks such as chemical reactions and radiation.

An important special case of the energy equation occurs in one-dimensional steady flow, see Fig. 3.4. The shear work done at the boundary is zero because the velocity is either zero or normal to the shear force. The work done by pressure,

cs

is only performed at the inflow and outflow boundaries where the flow velocity is not zero. If dissipation and chemical reactions are neglected, Eq. 3.25 reads