## Chapter 6

Viscous Flow in Ducts

## Differential Conservation Equations for Inviscid Flows

### Overview

Chapter 6 is devoted to the governing equations for compressible flows on differential form, i.e. it is a chapter very much like Chapter 2.

### Sections

#### 6.2 Differential Equations in Conservation Form

In this section the continuity, momentum, and energy equations on differential conservation form are derived. The starting point for the derivation is the integral form of the equations obtained in Chapter 2. The new set of equations constitutes a framework that can be applied and evaluated in any point of a three-dimensional inviscid flow.

$$\frac{\partial \rho}{\partial t} + \nabla \cdot(\rho{\mathbf{v}})=0$$ $$\frac{\partial}{\partial t}(\rho{\mathbf{v}}) + \nabla \cdot(\rho{\mathbf{vv}})+\nabla p = \rho{\mathbf{f}}$$ $$\frac{\partial}{\partial t}(\rho e_{o}) + \nabla \cdot(\rho h_{o}{\mathbf{v}}) = \rho({\mathbf{f}}\cdot{\mathbf{v}})$$

#### The Substantial Derivative

In this section the substantial derivative is introduced

$$\frac{D}{Dt}=\frac{\partial}{\partial t}+{\mathbf{v}}\cdot \nabla$$

... the time rate of change of any quantity associated with a particular moving fluid element is given by the substantial derivative ...

... the properties of the fluid element are changing as it moves past a point in a flow because the flowfield itself may be fluctuating with time (the local derivative) and because the fluid element is simply on its way to another point in the flowfield where the properties are different (the convective derivative) ...

#### Differential Equations in Nonconservation Form

The differential equations derived in this section are applicable when one wants to follow the development of a specific fluid element in space and time. A new set of governing equations are derived. The continutiy equation reads

$$\frac{D\rho}{Dt}+\rho(\nabla\cdot{\mathbf{v}})=0$$

... the mass of a fluid element made up of a fixed set of particles (molecules or atoms) is constant as the fluid element moves through space ...

The momentum equation:

$$\frac{D{\mathbf{v}}}{Dt}+\frac{1}{\rho}\nabla p={\mathbf{f}}$$

The energy equation:

$$\frac{De}{Dt}+\frac{p}{\rho}(\nabla\cdot{\mathbf{v}})=\dot{q}$$

In the last few pages of this section alternative formulations of the energy equation are derived. The total entropy formulation of the energy equation is analysed and discussed in some detail.

$$\frac{Dh_{o}}{Dt}=\frac{1}{\rho}\frac{\partial p}{\partial t}+\dot{q}+{\mathbf{f}}\cdot{\mathbf{v}}$$

As can be seen in the equation above, the total enthalpy of a moving fluid element in an inviscid flow can change due to

• unsteady flow: $$\partial p/\partial t \ne 0$$
• heat transfer: $$\dot{q} \ne 0$$
• body forces: $${\mathbf{f}}\cdot{\mathbf{v}} \ne 0$$

Adiabatic flow and without body forces gives:

$$\frac{Dh_{o}}{Dt}=\frac{1}{\rho}\frac{\partial p}{\partial t}$$

$$\frac{Dh_{o}}{Dt}=0$$

$$h_{o}$$ is constant along streamlines!

It should be noted that the equations derived in this sections are not non-conservative as such, the term non-conservative refers to properties of the equations when used for the development of CFD tools.

#### The Entropy Equation

From the first and second law of thermodynamics we have

$$\frac{De}{Dt}=T\frac{Ds}{Dt}-p\frac{D}{Dt}\left(\frac{1}{\rho}\right)$$

which is called the entropy equation. By comparing with the energy equation one can see that if the flow is adiabatic we get

$$\frac{Ds}{Dt}=0$$

which means that entropy is constant for moving fluid element. Furthermore, if the flow is steady we have

$$\frac{Ds}{Dt}=\frac{\partial s}{\partial t}+({\mathbf{v}}\cdot\nabla)s=({\mathbf{v}}\cdot\nabla)s=0$$

which implies that entropy is constant along streamlines.

#### Crocco's Theorem

... Crocco's theorem is a relation between gradients of total enthalpy, gradients of entropy, and flow rotation ...
$$T \nabla s=\nabla h_{o}+\frac{\partial {\mathbf{v}}}{\partial t}-{\mathbf{v}}\times(\nabla\times{\mathbf{v}})$$

Note: $$\nabla\times{\mathbf{v}}$$ is the vorticity of the fluid

... when a steady flow field has gradients of total enthalpy and/or entropy Crocco's theorem dramatically shows that it is rotational ...
• Entropy change over shock
• $$s$$ is constant upstream of shock
• jump in $$s$$ across shock depends on local shock angle
• $$s$$ will vary from streamline to streamline downstream of shock
• $$\nabla s\neq 0$$ downstream of shock
• Total enthalpy upstream of shock
• $$h_{o}$$ is constant along streamlines
• $$h_{o}$$ is uniform
• Total enthalpy downstream of shock
• $$h_{o}$$ is uniform
• $${\mathbf{v}}\times(\nabla\times{\mathbf{v}})\neq 0$$ downstream of a curved shock
• the rotation $$\nabla\times{\mathbf{v}}\neq 0$$ downstream of a curved shock

Explains why it is difficult to solve such problems by analytic means!

### Study Guide

The questions below are intended as a "study guide" and may be helpful when reading the text book.

1. How do we usually define the Reynolds number for pipe flows?
2. What does critical Reynolds number mean for a pipe flow?
3. What does entrance length mean? How is the flow velocity profile changed during the entrance length?
4. What does fully developed pipe flow mean?
5. Give three examples of sources of local losses in a pipe system
6. Start from Bernoullis extended equation and the momentum equation for a pipe with the length $$L$$ and diameter $$D=2R$$ and show that $$\Delta p_f=2\tau_w\dfrac{L}{R}$$
7. The Darcy friction factor is defined as $$f=\dfrac{8\tau_w}{\rho V^2}$$ Rewrite the friction loss from the previous question using the Darcy friction factor
8. For laminar pipe flows, show that: $$f=\dfrac{64}{Re_D}$$
9. What parameters effects the magnitude of $$f$$?
10. For fully developed laminar pipe flow, the velocity profile can be expressed as $$u=u_{max}\left(1-\dfrac{r^2}{R^2}\right)$$ Show that the average velocity in fully developed laminar pipe flow is half the maximum velocity.
11. Compare the velocity profiles for fully developed laminar and turbulent flow, which of the flows gives the highest wall shear stress for a given mass flow?
12. For fully developed turbulent pipe flow, the mean velocity profile can be approximated using the 1/7-rule $$u=u_{max}\left(\dfrac{r}{R}\right)^{1/7}$$ Why should this not be used directly for the calculation of wall shear stress?
13. Give three characteristics of turbulent flow
14. Turbulent flow is dissipative. What does that mean?
15. What defines the largest and smallest length scales in a turbulent flow?
16. Explain the concept of Reynolds decomposition
17. Why do one often want to use time-averaged equations when studying turbulent flow while that is not the case for laminar flows?
18. Explain the closure problem related to the Reynolds-averaged flow equations
19. In the Reynolds decomposition, the velocity components and pressure are divided into an average part and a fluctuating part as for example $$u=\bar{u}+u'$$ Define the time average and show that the time average of the fluctuating component is identically equal to zero.
20. How is the intensity of the fluctuating velocity component specified?
21. Derive the continuity equation for the time-averaged velocity field for incompressible turbulent flow starting from the general continuity equation on differential form $$\dfrac{\partial \rho}{\partial t}+\dfrac{\partial\left(\rho u\right)}{\partial x}+\dfrac{\partial\left(\rho v\right)}{\partial y}+\dfrac{\partial\left(\rho w\right)}{\partial z}$$
22. Derive the $$x$$-component of the Navier-Stokes equation for turbulent flow. Explain the physical meaning of each of the terms in the equation.
23. Show that the term $$-\rho \overline{u'v'}$$ can be interpreted as a shear stress
24. How can the turbulent shear stress be related to the mean flow using the turbulence viscosity $$\mu_t$$? What is this assumption called?
25. Define the friction velocity $$u^\ast$$
26. Derive the average velocity distribution in the viscous sublayer starting from $$\overline{u}\dfrac{\partial \overline{u}}{\partial x}+\overline{v}\dfrac{\partial \overline{v}}{\partial y}=\dfrac{1}{\rho}\dfrac{\partial \tau}{\partial y}$$
27. How does the turbulence viscosity $$\mu_t$$ compare to the fluid viscosity $$\mu$$ in the viscous sublayer and in the fully turbulent region, respectively?
28. How does the total shear stress vary with distance from the wall in the viscous sublayer and in the fully turbulent region, respectively?
29. How are the velocity profiles characterized mathematically in the viscous sublayer and in the fully turbulent region, respectively?
30. Make a schematic representation of the non-dimensional velocity $$u^+$$ as a function of the non-dimensional wall distance $$y^+$$ for a turbulent boundary layer. The velocity profile can be divided into different regions. Name these regions.
31. What relation for the local average velocity is used for the derivation of the friction factor $$f$$ for turbulent pipe flow?
32. The Moody chart is an engineering tool that can be used for estimation of pressure losses in a pipe flow
1. Why does the Moody chart not give reliable values in the Reynolds number range $$2000 < Re < 4000$$?
2. How does $$f$$ vary with the Reynolds number in the fully turbulent regime?
3. What is the effect of surface roughness on the friction factor?
4. What can we say about the Moody chart when it comes to accuracy?
33. How is the hydraulic diameter defined and how can it be used for calculation of the friction factor $$f$$ for laminar and turbulent flow in non-circular ducts?
34. Define the loss coefficient $$K$$

 Document Archive MTF053_C06.pdf Lecture notes chapter 6 MTF053_Turbulence.pdf Complementary material - Turbulent flow theory MTF053_Formulas-Tables-and-Graphs.pdf A collection of formulas, tables, and graphs