Reference Pages
Mechanical Engineering

    Scroll back to top

    Introduction

    In previous chapters, fluid mechanics problems were primarily analyzed by applying conservation laws to systems or control volumes. While these governing equations provide a complete mathematical description of fluid flow, many engineering problems involve several interacting variables such as velocity, pressure, density, viscosity, gravity, and geometry that combine in complex or nonlinear ways. As was seen, these effects make the formulation of analytic solutions extremely difficult. Moreover, as the number of variables increases, it becomes more difficult to determine which physical effects actually dominate the flow behavior.

    Dimensional analysis provides a systematic method for simplifying these problems by combining variables into dimensionless groups. Rather than studying each dimensional quantity independently, the flow may instead be described using parameters that compare competing physical effects, such as inertial, viscous, gravitational, or compressibility effects.

    A major tool used in this process is the Buckingham Pi Theorem, which states that a problem involving several dimensional variables may be rewritten in terms of a smaller set of independent dimensionless groups called \( \pi \) terms.These groups help reduce problem complexity, organize experimental data, and develop generalized relationships between variables.

    Dimensionless parameters also form the basis of similitude and scale modeling. In many engineering applications, full-scale systems are too expensive or impractical to test directly, so smaller laboratory models are used instead. If the correct similarity conditions are satisfied, the behavior of the model can be used to predict the behavior of the full-scale prototype.

    This chapter develops the Buckingham Pi Theorem, introduces common dimensionless groups used in fluid mechanics, discusses experimental correlations and scale models, and shows how important dimensionless parameters may also emerge directly from the governing equations through non-dimensionalization.

    Motivation

    In many fluid mechanics problems, the primary goal is to determine how one quantity depends on several others. For example, the drag force acting on an object moving through a fluid may depend on the fluid velocity, density, viscosity, and the size of the object. Symbolically, this relationship might be written as

    $$ F_D = f(V,\rho,\mu,L) $$

    While conservation laws and governing equations may fully describe the flow, solving those equations directly is often difficult or impractical for real engineering systems. In addition, experimentally studying every variable independently would require a large number of tests. Dimensional analysis helps simplify this process by combining variables into dimensionless groups. Rather than searching for a relationship between many dimensional quantities, the problem may instead be rewritten in terms of a smaller set of dimensionless parameters. This reduces problem complexity while still preserving the important physical behavior of the flow.

    Expressing results in dimensionless form is also useful because experimental data obtained from one system can often be applied to other similar systems. This idea forms the basis for experimental correlations, scale models, and similitude in fluid mechanics.

    Buckingham Pi Theorem

    The Buckingham Pi Theorem is a method for rewriting a dimensional physical relationship in terms of dimensionless groups. These dimensionless groups are called \( \pi \) terms. The main idea is that a problem with many dimensional variables can usually be described using fewer dimensionless parameters. It is then possible to conduct experiments that specify only the value of these dimensionless combination of variables as opposed to the individual value of each dimensional variable to identify the functional relationship between the dimensionless quantities. The Buckingham Pi Theorem can be overviewed as follows:

    Suppose a fluid mechanics problem depends on \( n \) dimensional variables:

    $$ q_1,\ q_2,\ q_3,\ldots,\ q_n $$
    If these variables contain \( r \) independent dimensions, such as mass \( M \), length \( L \), and time \( T \) (or \( F \),\( L \), and \( T \) as was emphasized in chapter 1), then the number of independent dimensionless groups needed to relate these variables is
    $$ n-r $$
    The original relationship
    $$ f(q_1,q_2,\ldots,q_n)=0 $$
    can then be rewritten as
    $$ F(\Pi_1,\Pi_2,\ldots,\Pi_{n-r})=0 $$
    or, solving for one group,
    $$ \Pi_1=\phi(\Pi_2,\Pi_3,\ldots,\Pi_{n-r}) $$
    where
    $$ \Pi_2,\Pi_3,\ldots,\Pi_{n-r} $$

    are the dimensionless quantities known as Pi terms. They are a repackaging of the dimensional variables.

    Note that the theorem does not give the exact function \( \phi \). In this way, dimensional analysis is not a method to solve or characterize the analytical relationship between variables. Instead, it identifies which dimensionless parameters the final relationship must depend on in a way that can greatly simplify empirical analysis.

    Visualization on dimensional analysis and the development of empirical correlations between non-dimensional groups

    Carrying Out the Buckingham Pi Theorem

    The method of repeating variables is a common method for determining Pi terms and applying the Buckingham Pi Theorem. It can usually be broken down
    1. List all variables expected to influence the problem.
    2. Write the dimensions of each variable using basic dimensions such as
      3. mass \( M \), length \( L \), time \( T \), and temperature \( \Theta \).
    3. Count the total number of variables, \( n \).
    4. Count the number of independent reference dimensions, \( r \).
    5. Determine the number of dimensionless groups using:
      6. $$ n-r $$
    6. Choose \( r \) repeating variables. These variables should collectively
      7. contain all fundamental dimensions and should not already form a dimensionless group by themselves.
    7. Combine each non-repeating variable with the repeating variables to
      8. form a \( \pi \) term. In general, a pi term takes the form:
      $$ \Pi_i = q(\text{repeating}_1)^a (\text{repeating}_2)^b (\text{repeating}_3)^c $$
      where \( q \) is a non-repeating variable and \( a,b,c \) are unknown exponents.
    8. Solve for the exponents that make each \( \pi \) term dimensionless by
      9. setting the powers of each fundamental dimension equal to zero.
    The repeating variables should be chosen carefully. They should collectively contain all of the basic dimensions in the problem, and they should not be able to combine to create a dimensionless group by themselves, that is, they should be dimensionally independent from one another.

    Example: Drag Force on a Body

    Consider the drag force acting on a body moving through a fluid. Assume the drag force depends on velocity, fluid density, fluid viscosity, and a characteristic length (Chapter 9 will demonstrate a more complete relationship for drag):
    $$ F_D = f(V,\rho,\mu,L) $$
    Step 1: List all variables expected to influence the problem. The variables are:
    $$ F_D,\ V,\ \rho,\ \mu,\ L $$
    Step 2: Write the dimensions of each variable using fundamental dimensions. Using mass \( M \), length \( L \), and time \( T \):
    $$ [F_D]=MLT^{-2} $$
    $$ [V]=LT^{-1} $$
    $$ [\rho]=ML^{-3} $$
    $$ [\mu]=ML^{-1}T^{-1} $$
    $$ [L]=L $$
    Step 3: Determine the number of variables, \( n \). There are five variables:
    $$ n=5 $$
    Step 4: Determine the number of independent reference dimensions, \( r \). The variables contain the dimensions:
    $$ M,\ L,\ T $$
    Thus,
    $$ r=3 $$
    Step 5: Determine the number of dimensionless groups. Using the Buckingham Pi Theorem:
    $$ n-r=5-3=2 $$
    Therefore, two independent dimensionless groups are required. Step 6: Choose \( r \) repeating variables. Choose:
    $$ \rho,\ V,\ L $$
    These variables collectively contain all fundamental dimensions and are dimensionally independent. Step 7: Form dimensionless products. The first non-repeating variable (variable remaining after our selection of repeating variables) is \( F_D \), so form:
    $$ \Pi_1 = F_D \rho^a V^b L^c $$
    The second non-repeating variable is \( \mu \), so form:
    $$ \Pi_2 = \mu \rho^a V^b L^c $$
    Step 8: Solve for the exponents required to eliminate dimensions. For the first pi term,
    $$ \Pi_1 = F_D \rho^a V^b L^c $$
    Substitute dimensions:
    $$ \Pi_1 = (MLT^{-2})(ML^{-3})^a(LT^{-1})^b(L)^c $$
    Collect powers of each dimension:
    $$ \Pi_1 = M^{1+a}L^{1-3a+b+c}T^{-2-b} $$
    For \( \Pi_1 \) to be dimensionless we can for the unknown exponents piece-wise for each dimension:
    $$ 1+a=0 $$
    $$ 1-3a+b+c=0 $$
    $$ -2-b=0 $$
    Solving:
    $$ a=-1,\qquad b=-2,\qquad c=-2 $$
    Thus,
    $$ \Pi_1 = \frac{F_D}{\rho V^2 L^2} $$
    Now repeat the process for the second pi term:
    $$ \Pi_2 = \mu \rho^a V^b L^c $$
    Substitute dimensions:
    $$ \Pi_2 = (ML^{-1}T^{-1})(ML^{-3})^a(LT^{-1})^b(L)^c $$
    Collect powers:
    $$ \Pi_2 = M^{1+a}L^{-1-3a+b+c}T^{-1-b} $$
    Set exponents equal to zero:
    $$ 1+a=0 $$
    $$ -1-3a+b+c=0 $$
    $$ -1-b=0 $$
    Solving:
    $$ a=-1,\qquad b=-1,\qquad c=-1 $$
    Therefore,
    $$ \Pi_2 = \frac{\mu}{\rho V L} $$
    This is actually just the inverse of the Reynolds number:
    $$ Re=\frac{\rho V L}{\mu} $$
    The final dimensionless relationship becomes:
    $$ \frac{F_D}{\rho V^2 L^2} = \phi(Re) $$
    This result shows that the dimensionless drag force depends on the Reynolds number.

    Finding Pi Terms by Inspection

    In some problems, dimensionless groups can be identified without explicitly solving for unknown exponents. This approach is called finding \( \pi \) terms by inspection. After gaining experience with common fluid mechanics problems and dimensionless parameters, this method is often significantly faster than the full repeating variable procedure.

    The basic idea is to look for combinations of variables that naturally form dimensionless ratios. Since the goal is to compare physical effects, many dimensionless groups arise from comparing quantities with similar units or similar physical meaning. For example:

    $$ \frac{\text{length}}{\text{length}}, \qquad \frac{\text{pressure}}{\text{pressure}}, \qquad \frac{\text{inertial effects}}{\text{viscous effects}} $$
    all produce dimensionless quantities. Common fluid mechanics dimensionless groups are often constructed by comparing:
    $$ \frac{\text{force}}{\text{inertial force scale}}, \qquad \frac{\text{pressure}}{\rho V^2}, \qquad \frac{\rho V L}{\mu} $$
    where \( \rho V^2 \) represents a dynamic pressure scale and
    $$ \frac{\rho V L}{\mu} $$
    is the Reynolds number.

    Example: Drag Force by Inspection

    Consider again the drag force acting on a body moving through a fluid:
    $$ F_D = f(V,\rho,\mu,L) $$
    There are five variables:
    $$ F_D,\ V,\ \rho,\ \mu,\ L $$
    and the reference dimensions are:
    $$ M,\ L,\ T $$
    Thus, the Buckingham Pi Theorem predicts:
    $$ n-r=5-3=2 $$
    Therefore, two independent dimensionless groups are required. The drag force has units of force, so a natural inertial force scale is based on:
    $$ \rho V^2 L^2 $$
    Since both quantities have dimensions of force, the first dimensionless group may be written directly as:
    $$ \Pi_1 = \frac{F_D}{\rho V^2 L^2} $$
    Next, viscosity is commonly compared against inertial effects using the Reynolds number:
    $$ \Pi_2 = Re = \frac{\rho V L}{\mu} $$
    Thus, by inspection, the drag relationship becomes:
    $$ \frac{F_D}{\rho V^2 L^2} = \phi(Re) $$
    This result is identical to the relationship obtained earlier using the method of repeating variables, but the dimensionless groups were identified directly without solving for unknown exponents. Essentially, we can just formulate pi terms directly by using intuition for how dimensionless quantities are formed. Over time, as one becomes more familiar with dimensional analysis, dimensionless quantities, or a specific set of variables, developing pi terms by inspection becomes much simpler than the method of repeating variables.

    Key Dimensionless Groups

    When working through the Buckingham Pi Theorem, one might notice that certian dimensionless quantities appear repeatedly. Indeed, many of these quantities aren't simple an convenient mathematical packaging, but rather carry with them physical significance. Many dimensionless groups appear repeatedly throughout fluid mechanics because they compare the relative importance of different physical effects. These parameters are useful for characterizing flow behavior, correlating experimental data, and establishing a connection between models of different scales or prototypes.

    Additionally, as was already seen in Chapter 6, they are particulary useful in identifying when certain simplifying assumptions are valid such as neglecting viscous effects or assuming incompressibility, the entire foundation in most introductory fluid mechanics courses. To that end, the Reynolds number, the most important and common dimensionless quantity incompressible flow and the mach number, both of which have been mentioned in previous chapters, are reviewed again here.

    Reynolds Number

    The Reynolds number compares the relative magnitude of inertial effects or forces in a flow to viscous effects:
    $$ Re=\frac{\rho V L}{\mu} =\frac{V L}{\nu} $$
    where:
    • \( \rho \) is fluid density,
    • \( V \) is a characteristic velocity,
    • \( L \) is a characteristic length,
    • \( \mu \) is dynamic viscosity,
    • \( \nu \) is kinematic viscosity.
    Large Reynolds numbers indicate flows dominated by inertial effects, while small Reynolds numbers indicate flows where viscous effects are important. The qualitative behavior of a flow is often highly depenedent on the Reynolds number and as has already been shown, the relative importance of viscosity versus inertial effects determines whether major simplifying assumptions are appropriate for the general Navier-Stokes equations. As will be expanded on in Chapter 7 as well, the Reynolds number is especially important in determining whether a flow is laminar or turbulent, an essential classification with impliciations for flow profiles, heat transfer, energy losses and more.

    Mach Number

    The Mach number compares flow velocity to the local speed of sound:
    $$ Ma=\frac{V}{c} $$
    where:
    • \( V \) is the flow velocity,
    • \( c \) is the local speed of sound.

    The Mach number measures the importance of compressibility effects within a flow For low Mach numbers, density changes are often negligible, while compressibility effects become increasingly important at higher Mach numbers. With this in mind, a typical incompressible flow analysis will usually not include the Mach number, however; as was stated in Chapter 1, it is vitally important to use the Mach number to consider whether an assumption about incompressibility is valid. Engineers who work on on high-speed flows or those involving transonic or supersonic flows such as those encountered in jet engines or turbines will see that many functional relationships between common flow variables become functions of the Mach number.

    The following table provides an overview of some common dimensionless groups encountered in fluid mechanics. Whether or not a specific dimensional quantity is relevant for a fluids problem depends heavily on the nature of the problem, the working fluid, and the relevant forces at play. Additionally, as the last two entries demonstrate, dimensionless quantities also appear quite frequently in other related fields, such as thermodynamics and heat transfer. Many energy transfer mechanisms, particularity those involving convention depend on both heat transfer properties of materials but also flow properties and are therefore often referred to as thermo-fluidics problems. Many of the principles developed here for incompressible flow, can be naturally extended into heat transfer and other adjacent fields.

    Notable dimensionless groups in fluid mechanics and heat transfer

    Name Symbol Definition Primary Comparison
    Reynolds Number \( Re \) \( \displaystyle Re=\frac{\rho V L}{\mu}=\frac{VL}{\nu} \) Inertial / viscous effects
    Mach Number \( Ma \) \( \displaystyle Ma=\frac{V}{c} \) Flow speed / speed of sound
    Froude Number \( Fr \) \( \displaystyle Fr=\frac{V}{\sqrt{gL}} \) Inertial / gravitational effects
    Euler Number \( Eu \) \( \displaystyle Eu=\frac{\Delta p}{\rho V^2} \) Pressure / inertial effects
    Weber Number \( We \) \( \displaystyle We=\frac{\rho V^2 L}{\sigma} \) Inertial / surface tension effects
    Strouhal Number \( St \) \( \displaystyle St=\frac{fL}{V} \) Unsteady / convective effects
    Prandtl Number \( Pr \) \( \displaystyle Pr=\frac{\nu}{\alpha} \) Momentum diffusion / thermal diffusion
    Nusselt Number \( Nu \) \( \displaystyle Nu=\frac{hL}{k} \) Convective / conductive heat transfer

    Scale Models

    In many engineering applications, directly testing a full-scale system may be too expensive, impractical, or difficult to control. Instead, smaller laboratory models are often constructed and tested under controlled conditions, such as aircraft testing in wind tunnels.

    The purpose of a scale model is not simply to create a smaller geometric copy of the real system, but rather to reproduce the important flow physics of the prototype. This is accomplished by matching the appropriate dimensionless groups between the model and the real system or in other words, equating pi terms. Given that a physical relationship is written in dimensionless form as:

    $$ \Pi_1=\phi(\Pi_2,\Pi_3,\ldots) $$
    If the relevant dimensionless groups are the same for both the model and the prototype,
    $$ (\Pi_2)_m=(\Pi_2)_p, \qquad (\Pi_3)_m=(\Pi_3)_p, \qquad \ldots $$
    then the remaining dimensionless term (which is usually constructed as the dependent variable) should also match:
    $$ (\Pi_1)_m=(\Pi_1)_p $$
    In practice this means that if one develops conditions for a scale model characterized by specific values for dimensionless variables, then one can use these results to predict the outcome for a similar full-scale system whose flow physics is defined by the same dimensionless parameters. For example, consider flow over a body where the drag relationship depends on the Reynolds number as before:
    $$ \frac{F_D}{\rho V^2L^2}=\phi(Re) $$
    To preserve similar flow behavior between the model and prototype, the Reynolds numbers should match:
    $$ (Re)_m=(Re)_p $$
    or
    $$ \left( \frac{\rho VL}{\mu} \right)_m = \left( \frac{\rho VL}{\mu} \right)_p $$
    If the same fluid is used in both cases, then density and viscosity are the same, giving:
    $$ V_mL_m=V_pL_p $$
    This means that a smaller model may require a larger velocity in order to maintain the same Reynolds number and preserve similar fluid behavior.
    Conditions for model similitude
    Most of the time, in developing pi terms and dimensionless quantities, one will notice that they fall into one of two categories. Oftentimes one of the pi terms will express some geometric result, such as the ratio of two length such as height and width for instance
    $$ \frac{h}{w} $$
    This shows that one part of establishing similitude between a laboratory and full-scale model is about establishing geometric similarity - equating the shape or profile a test subject. In the above example, it was also seen that a matching Reynolds number was required, indicating that we needed to equate flow conditions or physics as governed by flow variables and fluid properties. Thus a second category of similarity is the dynamic similarity. This usually bears out as matching the dimensionless values of the Reynolds number, Mach number, or Froude number. These two categories make sense; the laboratory model must recreate the same geometric features and flow physics as the corresponding full-scale prototype in order to make accurate predictions.

    Non-Dimensionalizing Equations

    Another important application of dimensional analysis is rewriting governing equations and exact solutions in dimensionless form. This process is called non-dimensionalization.

    The basic idea is to scale variables using characteristic reference quantities. For example, a velocity may be scaled by a characteristic speed \( V \), a length by a characteristic length \( L \), a time by a characteristic flow time \( L/V \), and a pressure by a characteristic pressure scale \( \rho V^2 \):

    $$ u^*=\frac{u}{V}, \qquad x^*=\frac{x}{L}, \qquad t^*=\frac{t}{L/V}, \qquad p^*=\frac{p}{\rho V^2} $$

    The starred quantities are dimensionless variables that have been normalized with respect to their reference quantities.

    Once the dimensionless variables are defined, they are substituted into the original governing equations. The equations are then algebraically rearranged so that all remaining terms are dimensionless. During this process, dimensionless groups naturally emerge from the mathematics itself. For example, the incompressible Navier--Stokes equations may be written as:

    $$ \rho \left( \frac{\partial \vec{V}}{\partial t} + (\vec{V}\cdot\nabla)\vec{V} \right) = -\nabla p + \mu\nabla^2\vec{V} $$
    After introducing characteristic scales and rewriting the equation in dimensionless form, the result becomes:
    $$ \frac{\partial \vec{V}^*}{\partial t^*} + (\vec{V}^*\cdot\nabla^*)\vec{V}^* = -\nabla^* p^* + \frac{1}{Re}\nabla^{*2}\vec{V}^* $$
    where
    $$ Re=\frac{\rho VL}{\mu} $$

    is the now familiar Reynolds number. The reciprocal of the Reynolds number now appears directly as the coefficient multiplying the viscous term. Large Reynolds numbers therefore correspond to flows where inertial effects dominate, while small Reynolds numbers indicate flows where viscous effects are more important, just as was discussed earlier.

    Dimensionless forms are also useful when rewriting exact solutions. For example, capillary rise (as was touched on in the introductory reference content) in a tube may be written as:

    $$ h=\frac{2\sigma\cos\theta}{\rho g R} $$
    Dividing by a characteristic height scale which is the height for perfect wetting or a contact angle of zero,
    $$ h_0=\frac{2\sigma}{\rho g R} $$
    gives the dimensionless form:
    $$ h^*=\frac{h}{h_0}=\cos\theta $$
    where
    $$ h^*=\frac{h}{h_0} $$

    is the dimensionless height.

    Expressing equations and solutions in dimensionless form generally doesn't chart new progress in solving an analytic equation. Nevertheless, it often simplifies the mathematics, highlights the dominant physics, and allows the results to apply to a broader class of similar flow problems.