Fluid Mechanics Chapter 8 Dimensional Analysis and Similitude Dr. Amer Khalil Ababneh Introduction Because of the complexity of fluid mechanics, the design of many fluid systems relies heavily on experimental results. Tests are typically carried out on a subscale model, and the results are extrapolated to the full-scale system (prototype). The principles underlying the correspondence between the model and the prototype are addressed in this chapter. Dimensional analysis is the process of grouping of variables into significant dimensionless groups, thus reducing problem complexity. Similitude (Similarity) is the process by which geometric and dynamic parameters are selected for the subscale model so that meaningful correspondence can be made to the full size prototype. 8.2 Buckingham Π Theorem In 1915 Buckingham showed that the number of independent dimensionless groups (dimensionless parameters) can be reduced from a set of variables in a given process is n - m, where n is the number of variables involved and m is the number of basic dimensions included in the variables. Buckingham referred to the dimensionless groups as Π, which is the reason the theorem is called the Π theorem. Henceforth dimensionless groups will be referred to as π-groups. If the equation describing a physical system has n dimensional variables and is expressed as then it can be rearranged and expressed in terms of (n - m) π- groups as 1 ( 2 , 3 ,..., nm ) Example If there are five variables (F, V, ρ, μ, and D) to describe the drag on a sphere and three basic dimensions (L, M, and T) are involved. By the Buckingham Π theorem there will be 5 - 3 = 2 π-groups that can be used to correlate experimental results in the form F= f(V, r, m, D) 8.3 Dimensional Analysis Dimensional analysis is the process used to obtain the π-groups. There are two methods: the step-by-step method and the exponent method. The Step-by-Step Method The method is best describe by an example EXAMPLE 8.1 Π-GROUPS FOR BODY FALLING IN A VACUUM There are three significant variables for a body falling in a vacuum (no viscous effects): the velocity V; the acceleration due to gravity, g; and the distance through which the body falls, h. Find the π-groups using the step-by-step method. Solution The variables involved in this example are n=3, V, g, and h. The dimension of these variables in terms of the basic dimensions are: ►Note the notation used: brackets means dimension of the variable contained between the brackets. ►The number of basic dimension that appear in the dimension of the variable is 2 ►Hence, the number of -groups is n – m = 3 – 2 = 1 ►Construct the following table by listing the variable along with their dimensions in terms of the basic dimensions. The steps involved in this table are 1- List the variable along with their dimensions in the first and second columns. 2- Choose a variable to combine new variables with and to eliminate a dimension from the new formed variables. List these in the second and third column 3- If the resulting dimensions is 0 (dimensionless) stop 4- Else, choose another variable from the third column to form new variables and to eliminate that dimension. 5- Repeat as necessary to arrive at dimensionless groups. The final result as expected one p-group, Consequently, the functional form is EXAMPLE 8.2 Π-GROUPS FOR DRAG ON A SPHERE USING STEP-BY-STEP METHOD The drag FD of a sphere in a fluid flowing past the sphere is a function of the viscosity μ, the mass density ρ, the velocity of flow V, and the diameter of the sphere D. Use the step-by-step method to find the π-groups. Solution Given FD = f(V, ρ, μ D). Dimensions of significant variables, Follow the same steps as in previous example and construct table as shown on next slide. The final -groups are Thus, the final functional form is 1 = f(2) The Exponent Method An alternative method for finding the π-groups is the exponent method. This method involves solving a set of algebraic equations to satisfy dimensional homogeneity. The procedural steps for the exponent method is illustrated in the following example. EXAMPLE 8.3 Π-GROUPS FOR DRAG ON A SPHERE USING EXPONENT METHOD The drag of a sphere, FD, in a flowing fluid is a function of the velocity V, the fluid density ρ the fluid viscosity μ and the sphere diameter D. Find the π-groups using the exponent method. Solution Given FD = f(V, ρ, μ D). Dimensions of significant variables are Number of π-groups is 5 - 3 = 2. Choose repeating variables equal to the number of basic dimensions, m = 3. The repeating variables are typically r, V, D, Form product with the remaining dimensions at a time. Start with dimensions of force as follow, Dimensional homogeneity. Equate powers of dimensions on each side. For M, the equation is M: 1 + b = 0, implies b = -1 T: -2 – a = 0, implies a = -2 L: 1 + a -3b + c = 0, implies c = -2 Thus, the resulting -groups is: Repeat the same procedure with the viscosity as below, Solving for the exponents lead to the second -groups, The final functional form: 8.4 Common π-Groups The most common π-groups can be found by applying dimensional analysis to all the variables that might be significant in a general flow situation. Variables that have significance in a general flow field are the velocity V, the density ρ, the viscosity μ, and the acceleration due to gravity g. In addition, if fluid compressibility were likely, then the bulk modulus of elasticity, Ev, should be included. If there is a liquid-gas interface, the surface tension effects may also be significant. Finally the flow field will be affected by a general length, L, such as the width of a building or the diameter of a pipe. These variables will be regarded as the independent variables. The primary dimensions of the significant independent variables are: There are several other independent variables that could be identified for thermal effects, such as temperature, specific heat, and thermal conductivity. Inclusion of these variables is beyond the scope of this text. Typically one is interested in pressure distributions (p), shear stress distributions (τ), and forces on surfaces and objects (F) in the flow field. These will be identified as the dependent variables. The primary dimensions of the dependent variables are There are 10 significant variables, which, by application of the Buckingham Π theorem, means there are seven π-groups. Utilizing either the step-by-step method or the exponent method وyields The first three groups, the dependent π-groups, are identified by specific names. For these groups it is common practice to use the kinetic pressure, ρV2/2, instead of ρ V2. In most applications one is concerned with a pressure difference, so the pressure π- group is expressed as where Cp is called the pressure coefficient and p0is a reference pressure. The π-group associated with shear stress is called the shear-stress coefficient and defined as where the subscript f denotes “friction.” The π-group associated with force is referred to, here, as a force coefficient and defined as The independent π-groups are named after earlier contributors to fluid mechanics. The π-group VLρ/μ is called the Reynolds number, after Osborne Reynolds, and designated by Re. The group is rewritten as (V/c), since is the speed of sound, c. This π-group is called the Mach number and designated by M. The π-group ρLV2/σ is called the Weber number and designated by We. The remaining π-group is usually expressed as and identified as the Froude (rhymes with “food”) number * and written as Fr. The general functional form for all the π-groups is which means that either of the three dependent π-groups are functions of the four independent π-groups; that is, the pressure coefficient, the shear-stress coefficient, or the force coefficient are functions of the Reynolds number, Mach number, Weber number, and Froude number. Each independent π-group has an important interpretation as indicated by the ratio column in Table 8.3 (see textbook). The Reynolds number can be viewed as the ratio of kinetic to viscous forces. The kinetic forces are the forces associated with fluid motion. The Bernoulli equation indicates that the pressure difference required to bring a moving fluid to rest is the kinetic 2 pressure, ρV /2, so the kinetic forces, Fk should be proportional to The shear force due to viscous effects, Fν, is proportional to the shear stress and area and the shear stress is proportional to so Fν ~ μ VL. Taking the ratio of the kinetic to the viscous forces yields the Reynolds number. The magnitude of the Reynolds number provides important information about the flow. A low Reynolds number implies viscous effects are important; a high Reynolds number implies kinetic forces predominate. The Reynolds number is one of the most widely used π-groups in fluid mechanics. It is also often written using kinematic viscosity, Re = ρVL/μ = VL/ν. The other p-groups are also given physical interpretation: ►The Mach number is an indicator of how important compressibility effects are in a fluid flow ►The Froude number is important when gravitational force influences the pattern of flow, such as in flow over a spillway ►The Weber number is important in liquid atomization where surface tension of the liquid at the droplet’s surface is responsible for maintaining the droplet's shape 8.5 Similitude Scope of Similitude Similitude is the theory and art of predicting prototype performance from model observations.