Nondimensionalization

5 Key excerpts on "Nondimensionalization"

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  Modeling in Materials Processing

  • Jonathan A. Dantzig, Charles L. Tucker(Authors)
  • 2001(Publication Date)
  • Cambridge University Press

    (Publisher)

  78 SCALING AND MODEL SIMPLIFICATION 3.5 Nondimensionalization FOR NUMERICAL SOLUTIONS (ADVANCED) In practice, it is quite common to use a computer code to solve the equations of a pro-cess model numerically. Numerical solutions are attractive because they can retain all of the terms in the governing equations, treat complicated geometries, and incorpo-rate material properties that vary as a function of temperature, strain rate, and so on. When these codes are used, it is usually best to enter the data in dimensionless form. Well-written software for fluid flow, heat transfer, structural analysis, and the like does not make any assumptions about units for the data. Rather, it relies on the user to enter the data in a consistent set of units, and to interpret the results accordingly. For example, suppose we have a code that solves fluid flow problems. If the coordinates of the grid points are entered in units of meters, the density is entered in kilograms per cubic meter, and the viscosity in (newtons times seconds) per square meter, then the calculated velocities will be in meters per second and the calculated stresses will be in newtons per square meter. Although some simple problems can be solved numerically by entering the data values in some familiar unit system, in other cases this will create problems with roundoff errors and convergence. As a user, you can help a code do its best by keeping all of the values it works with close to unity. This sounds like scaling the variables and governing equations is exactly what is needed, but in fact there are a few important differences. For this reason we speak of Nondimensionalization for numerical solutions, to distinguish this activity from scaling analysis. 3.5.1 ISOTROPIC LENGTH SCALES In scaling analysis we are free to choose different length scales along different coor-dinate axes, and in fact this may allow us to eliminate some terms from the governing equations.

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  Incompressible Flow

  • Ronald L. Panton(Author)
  • 2013(Publication Date)
  • Wiley

    (Publisher)

  Ideally, fluid dynamic events on the hydraulic model, the computer model, and the real river should all agree. This means that running the hydraulic model at the conditions desired can check empirical coefficients in the computer model. In this way extreme conditions, which may never actually occur, can be verified. 8.10 NONDIMENSIONAL FORMULATION OF PHYSICAL PROBLEMS In many instances we know the equations that govern a problem and can write out the relevant laws and conditions. The fact that solutions of physical problems must be dimen- sionally homogeneous is only contained implicitly in the governing equations. It is often ignored as one finds the solution. If we recast the problem into nondimensional variables, we explicitly use the information that physical functions are dimensionally homogeneous. Boundary conditions and physical constants are used to nondimensionalize the dependent and independent variables. The nondimensional form of the problem will contain all the 8.10 Nondimensional Formulation of Physical Problems 175 necessary variables. Inspection of these equations will reveal the nondimensional functions without using the pi theorem. Moreover, frequently there is information contained in the governing equations that reduces the number of nondimensional variables even more than the pi theorem would predict. The advantages of nondimensonalizing a problem are great; the problem has the fewest variables and the simplest mathematical structure when expressed in nondimensional variables. Nondimensional variables may be thought of as variables whose scales or units of measurement come from the problem itself. In this sense they are natural scales. The standard units of measurement, such as the meter, the kilogram, and the second, have no special importance to any physical processes. The important scales (the S values of Eq. 8.7.4) come from the problem itself. Consider for a moment the anatomy of a nondimensional variable y ∗ .

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  Introduction to Convective Heat Transfer

  A Software-Based Approach Using Maple and MATLAB

  • Nevzat Onur(Author)
  • 2023(Publication Date)
  • Wiley

    (Publisher)

  The term-by-term Nondimensionalization of these governing differential equations leads directly to the related dimension- less numbers. We nondimensionalize each variable with proper, constant reference quantities. This method is not useful if the problem cannot be formulated in terms of governing differential equations. The method is also discussed in [6] and will be explained by examples. We will consider only incompressible fluids. Example 5.11 We wish to study the two-dimensional boundary layer problem over a flat plate as shown in Figure 5.E11. We assume a steady, incompressible, constant property flow over the flat plate with a constant free stream velocity U ∞ and temper- ature T ∞ . The pressure gradient is absent since U ∞ = constant. The flow at the leading edge of the plate has a uniform velocity U ∞ and temperature T ∞ . The plate length is L, and it is subject to constant wall temperature T w . The fluid density is ρ and the dynamic viscosity is μ. The velocity components in the x- and y-directions are u and v, respectively. Gravity force and viscous dissipation are neglected. We are interested in determining any dimensionless parameters that arise. x y L U ∞ U ∞ T ∞ T w Figure 5.E11 Problem description for Example 5.11. Solution For the present problem, the equations of momentum and energy are uncoupled, and they can be solved separately. Incompressible constant property flow: Velocity problem Invoking the boundary layer approximations, two-dimensional forms of the basic equations of mass and momentum are given below. The conservation of mass u x + v y = 0. The momentum equation ρ ( u u x + v u y ) = μ  2 u y 2 . The pressure gradient dp/dx = 0 for constant free stream velocity U ∞ . The boundary conditions are u(x, 0) = 0 v(x, 0) = 0 5.3 Nondimensionalization of Basic Differential Equations 117 u(x, ∞) = U ∞ u(0, y) = U ∞ . We will put the governing equations in dimensionless form.

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  Fluid Mechanics of the Atmosphere

  • Robert A. Brown(Author)
  • 1991(Publication Date)
  • Academic Press

    (Publisher)

  We have seen that if a nondimensional coefficient multiplying a term in the equation is known to be extremely small, then that particular dimensional term is small with respect to the other terms. This term is a candidate to be neglected. This could be formally done in an asymptotic expansion with respect to the small parameter. In this way terms are dropped from the com-plete equations and more easily solvable equations may result. There is a danger in this process, however, that is associated with the characteristic values chosen. Because if the characteristic value is chosen incorrectly, the resulting equations may not be valid, as in a domain where the characteristic value chosen wasn't typical of the dependent variable. 2 See the theory of Kaplun, S. (1967). Fluid Mechanics and Singular Perturbations, -a collection of papers. Academic Press, New York; Van Dyke, M. (1964). Perturbation Meth-ods in Fluid Mechanics, Applied Mathematics and Mechanics series, Vol. 8, Academic Press, New York.) 3.3 Similarity 155 Example 3.11 Consider the equation for one-dimensional flow along a streamline in the x-direction, which we can take at this point as given to be pip + gz + ~U2 = C = Po/p (3.16) where the pressure p = Po at z = 0 and u = O. Incompressible flow is assumed, with Po/p = 600 m 2/sec 2 • The region of interest is 100:Sz:s1IOm and 10 :s u :S U max = 30 m/sec Nondimensionalize the equation and discuss it. Solution From the given equation and boundary conditions, choose Po, Az (= 10 m), and U max (= 30 m/sec) for nondirnensionalizing p, z, and u. We let p' = p/P o , z' == z//).z, and u' = «io.; and substitute (in this case, the primed variables are ND) (Po/p)p' + gHz' + ~U~ u,z = C = Po/p (~.17) We are interested in the flow situation and therefore definitely want to retain the term involving the velocity.

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  Handbook of Fluid Dynamics

  • Richard W. Johnson(Author)
  • 2016(Publication Date)
  • CRC Press

    (Publisher)

  Dimensional analysis is most useful in complicated situa-tions. However, if the situation is so simple that one can write the governing physical laws (see Section 5.2), then the Pi theorem can Handbook of Fluid Dynamics 5 -34 be skipped and the equations and relations can be placed directly in nondimensional form. Often, because the governing laws contain more information, the result is sharper than simple dimensional analysis. Organizing experimental data also benefits from dimen-sional analysis. A discussion of this is given in Panton (2010). 5.8 Internal Incompressible Viscous Flow William S. Janna 5.8.1 Introduction The study of flow in closed conduits is extremely important because there are numerous engineering applications: water is transported about a household through tubes; petroleum prod-ucts are conveyed about a refinery through pipes; steam and condensed water are transported around a power plant through pipes and copper tubing. It is necessary in these applications to be able to mathematically predict the performance of a system so that it can be properly designed. The mathematics involved in modeling piping systems is rather classical. The objective in such a model is to relate the pressure drop experienced by the fluid to the fluid properties (density and viscosity), to certain flow parameters (volume flow rate or velocity), and to the geometry of the system (character-istic length for the duct cross section or its cross-sectional area). The pressure drop is related to a friction factor term, which in turn is related to the Reynolds number of the flow and the sur-face roughness of the duct material. Traditionally, this relationship is shown as a log-log graph of friction factor versus Reynolds number. Such a chart is called a Stanton diagram . Blasius (1913) was the first to correlate the fric-tion factor with the Reynolds number for smooth pipes.

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