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Topic: Nondimensionalization

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	Nondimensionalization - Wikipedia, the free encyclopedia
		Nondimensionalization is the partial or full removal of units from a mathematical equation by a suitable substitution of variables.
		Nondimensionalization is not the same as converting extensive quantities in an equation to intensive quantities, since the latter procedure results in variables that still carry units.
		The nondimensionalization factor for the energy is the same as the ground state of the harmonic oscillator.
	en.wikipedia.org /wiki/Nondimensionalization (1651 words)

	Nondimensionalization
		Nondimensionalization reduces the dependence of all aerodynamic forces and moments on density and airspeed to a simple factor.
		The nondimensional forces and moments are defined by Equations 44-49.
		The angle of attack and angle of sideslip, defined by Equations 5 and 6, are the nondimensional replacements for the dimensional wind-relative velocity components (
	www.aerojockey.com /papers/meng/node24.html (121 words)

	LapAppl::EmitNamelist
		If a nondimensionalization factor is active, that factor is multiplicatively applied to the numeric value before the value is emitted to the building namelist.
		This application of the nondimensionalization mechanism is useful for converting angles in radians to angles in degrees, the angular measurement standard used by Lapin.
		This application of the nondimensionalization mechanism is useful for converting mass in slugs to mass in pounds mass, the mass measurement standard used by Lapin.
	www.grc.nasa.gov /WWW/price000/lap/htm/lapappl_emitnamelist.html (3299 words)

	untitled
		We can similarly nondimensionalize the angular rates, P, Q, and R. This is also done to reduce the dependence of the aerodynamic forces and moments to simpler forms and to restrict the range of typical numerical values of these parameters.
		A dimensionless roll rate of 0.1, for example means that the tip of the airplane develops a normal velocity of 10% of the freestream speed (a change of more than 11 degrees in local angle of attack between the left and right tips).
		It is usually better to keep the nonlinear equations in their dimensional form, using the dimensionless expressions for the aerodynamic forces to then assemble the overall forces and moments in dimensional terms, and finally plug them into the full dimensional nonlinear EOMs.
	adg.stanford.edu /aa208/dynamics/nondimen.html (863 words)

	NonDimensionalization (Site not responding. Last check: 2007-10-20)
		Once you have picked a length scale Ls (1 m for example) and a force scale Fs (1 kN for example) then nondimensional quantities are calculated by dividing by combinations of the scales that remove all the units.
		Nondimensional quantities are typically denoted with a * on the symbol.
		Numbers that are calculated by Truss Solver are then also nondimensional with the same scales.
	www.aem.umn.edu /people/faculty/shield/software/truss/nondim.htm (465 words)

	Re: Nondimensionalization in FIDAP
		Nondimensionalization in FIDAP - Andrew Ward, Sun, 23 Jun 2002, 5:38 p.m.
		Re: Nondimensionalization in FIDAP - Jin-Wook LEE, Sun, 23 Jun 2002, 6:14 p.m.
		Re: Nondimensionalization in FIDAP - Andrew Ward, Sun, 23 Jun 2002, 9:37 p.m.
	www.cfd-online.com /Forum/fluent_archive_2002.cgi/read/8443 (199 words)

	AE 3005 Handout # 3
		Intuition and experience are needed to realize that this nondimensional parameter, called Reynolds number, is also a useful physical parameter.
		This parameter, while nondimensional, is not a significant parameter in incompressible flows, as experience shows.
		In such flows, viscous forces are small and may be neglected over most of the flow, except in very small regions called boundary layers over the airfoil.
	www.adl.gatech.edu /classes/shearflo/shear3/nondim.html (685 words)

	Re: Nondimensionalization
		There are two reasons to use a nondimensional form: i) That way it is easier to compare numerical and real life experiments, as it is now just a matter of scaling.
		The nondimensionalization should leave you with quantities which are all O(1).
		Re: Nondimensionalization - Tony, Fri, 29 Aug 2003, 5:40 p.m.
	www.cfd-online.com /Forum/main_archive_2003.cgi/read/28113 (222 words)

	Preconditioning the Helmholtz Equation for a 2-D Duct
		The governing differential equation for the propagation of sound in nondimensionalized form is
		The spatial unit for nondimensionalization is the width (y dimension) of the duct; the temporal unit for nondimensionalization is the width of the duct divided by the speed of sound; and the unit for nondimensionalization of transient potential is the width of the duct multiplied by the speed of sound.
		A Semi-Infinite Rectangular Duct is overlaid with Cartesian coordinates that are nondimensionalized in that lengths are expressed in units of the duct width along the y axis.
	www.nasatech.com /Briefs/Mar00/LEW16692.html (681 words)

	Governing Equations
		In the process of nondimensionalizing these equations, Taylor produces a length scale which becomes a repeated length scale in all of the work on thin fluid sheets.
		However, Taylor's discussion of the nondimensionalization leaves much to be desired and will be discussed here in more detail so it becomes more evident how the length scale is determined.
		The Bernoulli's equation, nondimensionalized with the newly defined length scale becomes Taylor's equation I5
	www.deas.harvard.edu /brenner/taylor/handouts/waterbell/node2.html (439 words)

	Finite Difference Equations
		Advection is effected using centered differencing in flux form, except at the upper boundary where an upwind scheme is implemented.
		The (nondimensional) hydrostatic equation is differenced as follows:
		These velocities are used to advance the temperature (and salinity) fields one timestep, and the process is repeated.
	www.es.ucsc.edu /~rop/pgman/node4.html (543 words)

	[No title] (Site not responding. Last check: 2007-10-20)
		The turbulent viscosity (mu_t) is nondimensionalized by the molecular viscosity at the reference temperature computed using Sutherland's law.
		The kinematic turbulent viscosity (nu_t) is of course nondimensionalized by mu_ref/rho_ref, where rho_ref is determined from the reference pressure and temperature.
		We will be releasing a new User's Manual with Version 2.0 which should be better organized and clearer than the original TR.
	www.arnold.af.mil /nparc/bb/msgAAAa11803.html (61 words)

	Exploring Complexity in Some Simple Nonlinear Chemical Kinetic Schemes -- from Mathematica Information Center
		These schemes model situations where precursor decay is neglected and included respectively and can represent both isothermal and thermokinetic processes.
		We make use of a consistent nondimensionalization that has the advantage of unifying all the previously published related models.
		A systematic investigation of the dynamical behavior within a subspace of the full parameter space reveals clearly distinguished regions where sequences of period doubling, chaos and mixed-mode oscillations exist.
	library.wolfram.com /infocenter/Articles/961 (209 words)

	Physical constant - Wikipedia, the free encyclopedia
		Ratios of like-dimensioned physical constants do not depend on unit systems in this way (the units cancel), so they are pure (dimensionless) numbers whose values a future theory of physics could conceivably hope to predict.
		Additionally, all equations describing laws of physics can be expressed without dimensionful physical constants via a process known as nondimensionalization, but the dimensionless constants will remain.
		Thus, theoretical physicists tend to regard these dimensionless quantities as fundamental physical constants.
	en.wikipedia.org /wiki/Physical_constants (1008 words)

	CiteULike: The Quasi-Steady-State Assumption: A Case Study in Perturbation (Site not responding. Last check: 2007-10-20)
		These estimates lead to a dimensionless formulation containing a different small parameter from that which is customary.
		Earlier results on the QSSA are surveyed in the context of the new nondimensionalization.
		It is shown that more comprehensive and simpler error estimates can now be made.
	www.citeulike.org /user/rklancer/article/224765 (242 words)

	dimensionless.nb (Site not responding. Last check: 2007-10-20)
		We can see how a typical nondimensionalization for the Navier-Stokes Equations would turn out.
		We substitute the Reynolds number into the equation.
		Note that the Reynolds number, R U ρ/μ is multiplying the inertia (nonlinear) term and the time derivative term in this nondimensional equation.
	www.nd.edu /~mjm/dimensionless/Links/dimensionless_lnk_8.html (67 words)

	ANS : Store : Electronic Articles
		In order to validate the numerical results over a wide range of parameters, an experimental test facility has been designed and constructed to simulate the hydrodynamics of downward-facing porous wetted walls.
		Nondimensionalization of the model shows that water can be adequately used as a simulant to validate the numerical results.
		Preliminary experimental results show good agreement with model predictions.
	www.ans.org /store/index.cgi?i=E100000-fst-43-3-366-377 (267 words)

	Microsoft PowerPoint Presentation (Site not responding. Last check: 2007-10-20)
		Nondimensionalization in Free Convection: What Velocity to Use?
		Nondimensional Navier-Stokes for Free Convection - Grashof Number
		Nusselt Number in Free Convection is a Function of Pr, Gr, L/D
	www.chem.mtu.edu /~fmorriso/cm310/heat_12_lecture_old/index.htm (56 words)

	[No title]
		The flow solver employs following scheme for the nondimensionalization of various parameters of interest.
		Considering the convention used for the coordinate system in USM3Dns and the nondimensionalization scheme described above, the freestream and other relevant quantities assume the following nondimensional values/form.
		Sutherland's law for the molecular viscosity mentioned above is given by:
	aaac.larc.nasa.gov /tsab/usm3d/OVERVIEW.html (206 words)

	Math 4803 Cover Page (Site not responding. Last check: 2007-10-20)
		1/12 finish van der Pol/start nondimensionalization and scaling
		1/14 nondimensionalization and scaling/start some perturbation theory and asymptotics
		Note: The perturbation theory stuff will come from lecture notes on nonlinear vibrations by Richard Rand and parts of the modeling book by Lin and Segal.
	www.math.gatech.edu /~mason/class/ma4803/cal.html (584 words)

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