Chapter 5

Dimensional Analysis and Similarity

table of content

Quasi-One-Dimensional Flow


Chapter 5 adds to the theory presented in Chapter 3 by extending the equations from on-dimensional flow to quasi one-dimensional flows. Quasi one dimensional implies that we still have variations of flow quantities in one direction only but we allow the cross section area of stream tubes to vary along the same direction as well. This means that we will be able to analyze axisymmetric channels with varying cross section areas such as for example convergent-divergent nozzles. One thing that is important to note is that for the equations to be valid the variation in cross section area has to be smooth.


5.2 Governing Equations

The governing equations for quasi one-dimensional flows are derived. The resulting continuity and energy equations are unchanged compared to the corresponding equations derived for one-dimensional flows in Chapter 3. In the momentum equation, however, we get a contribution from the pressure forces on the control volume perimeter surface (the surface aligned with the stream lines) that was not present in the momentum equation for one-dimensional flows. Moreover, the new term appears as an integral in the equation, which means that we need to rewrite the equations on differential form. This leads us to a momentum equation on a form known as the Euler equation.

$$ dp=-\rho udu $$
Quasi-One-Dimensional flow
Figure 1: In quasi-one-dimensional flows the stream tube cross section area is allowed to vary in one direction \(A=A(x)\)

5.3 Area-Velocity Relation

Starting from the continuity equation, the momentum equation, and making an assumption of isentropic flow, the area-velocity relation is derived. This relation is an important contribution to the compressible flow engineering field since it describes gives us information on how to accelerate a flow to supersonic speeds.

$$ \frac{dA}{A}=(M^2-1)\frac{du}{u} $$

5.4 Nozzles

In this section another important relation is derived, namely the area-Mach number relation. The area-Mach number relation gives the ratio of local area to throat area as a function of Mach number (or the other way around). The area-Mach number relation is valid for isentropic flows (i.e. no shocks allowed) and calorically perfect gases.

Area-Mach number relation
Figure 2: The area-Mach-number relation

With the area distribution of a nozzle given we will be able to get the Mach number as function of \(x\) and thus also (using the isentropic relations) \(p/p_o\), \(T/T_o\), etc. It should be noted that there are a infinite number of possible scenarios with isentropic subsonic flow through the entire nozzle but only one possible solution where we get supersonic flow in the divergent part of the nozzle. The nozzle flow is highly dependent on the back pressure. If the back pressure is to high, we will get a normal shock at an axial location in the divergent part of the nozzle. Upstream of the normal shock, the flow will follow the isenttropic supersonic solution and on the downstream side we will get subsonic flow. The normal shock moves downstream the nozzle as the back pressure is reduced and for one specific back pressure we will have a normal shock standing at the nozzle exit plane. This means that we will have isentropic shock-free supersonic flow throughout the divergent part pf the nozzle. Reducing the back pressure further, results in overexpanded conditions leading to generation of oblique shocks at the nozzle exit going out from the nozzle. With further decrease of back pressure we may reach pressure matched conditions (flow without shocks or expansion waves) and if we continue to decrease the back pressure after that we will get formation of expansion waves at the nozzle exit. In the latter case the nozzle is said to be operating at underexpanded conditions.

A relation for calculation of the mass flow through a choked nozzle is derived.

$$ \dot{m}=\frac{p_o A^*}{\sqrt{T_o}}\sqrt{\frac{\gamma}{R}\left(\frac{2}{\gamma+1}\right)^{(\gamma+1)/(\gamma-1)}} $$

Nozzle pressure ratio
Internal normal shocks
Figure 3: (a) Mach number as function of axial coordinate for different nozzle pressure ratios. (b) Normal shock location for different pressure ratios

under and over expanded nozzle flow
Figure 4: Underexpanded and overexpanded nozzle flow

5.5 Diffusers

Diffuser flow is exemplified by describing the basic concepts of a supersonic wind tunnel. As a first example, supersonic flow coming from a convergent-divergent nozzle is going through a normal shock and then a simple diffuser. This setup gives a significant improvement compared to let the flow from the nozzle directly to the atmosphere. A system of oblique shocks in a diffuser section of a wind tunnel results in way better performance.

5.6 Wave Reflection from a Free Boundary

Waves emanating from non-perfectly expanded nozzles will reflect at the jet shear layer and the jet centerline resulting in a shock diamond pattern downstream of the nozzle exit. The shock diamond flow is not governed by the quasi one-dimensional theory but still included in this section since it is associated with supersonic nozzle flows.

Shock reflection at free boundary
Expansion wave reflection at free boundary
Figure 5: Wave reflection at a free boundary (a) Shock reflection (b) expansion wave reflection

Study Guide

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

  1. If you are going to do an experimental investigation of a problem including a number of important physical variables, why is it beneficial to divide the variables into non-dimensional groups?
  2. Rewrite the continuity equation and the \(x\)-component of the momentum equation in non-dimensional form $$\dfrac{\partial u}{\partial x}+\dfrac{\partial v}{\partial y}+\dfrac{\partial w}{\partial z}=0$$ $$\dfrac{\partial u}{\partial t}+u\dfrac{\partial u}{\partial x}+v\dfrac{\partial u}{\partial y}+w\dfrac{\partial u}{\partial z}=g_x-\dfrac{1}{\rho}\dfrac{\partial p}{\partial x}+\dfrac{\mu}{\rho}\left(\dfrac{\partial^2u}{\partial x^2}+\dfrac{\partial^2u}{\partial y^2}+\dfrac{\partial^2u}{\partial z^2}\right)$$
  3. The flow resistance \(F_D\) of a body immersed in a flow can be divided into pressure forces and friction forces (MTF053 Dimensional-Analysis-and-Similarity.pdf)
    1. Show using Reynolds similarity that the pressure forces can be written as $$F_{D_p}=C_{D_p}(Re)A_p\dfrac{\rho U^2}{2}$$ where the drag coefficient \(C_{D_p}\) is a function of the Reynolds number only and \(A_p\) is the projected area of the body.
    2. Show using Reynolds similarity that the pressure forces can be written as $$F_{D_f}=C_{D_f}(Re)A_p\dfrac{\rho U^2}{2}$$ where the drag coefficient \(C_{D_f}\) is a function of the Reynolds number only and \(A_p\) is the projected area of the body.
  4. Explain the concepts geometric similarity, kinematic similarity, and dynamic similarity

Document Archive
MTF053_C05.pdf Lecture notes chapter 5
MTF053_Dimensional-Analysis-and-Similarity.pdf Complementaory material
MTF053_Formulas-Tables-and-Graphs.pdf A collection of formulas, tables, and graphs