H.N. Abramson - The dynamic behavior of liquids in moving containers. With applications to space vehicle technology (798543), страница 29
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Consequently, the most important part of an experimental design is the judgment of the designeras to which parameters, fluid properties, andsystem variables are important and which arenot. I n all cases, theory and previous experimental observations are combined in this judgment to the extent possible. However, if thesimulation is necessary a t all, some aspects ofthe underljkg theory must be speculative, unconfirmed, missing, or perhaps too difficult toevaluate.
The methods implied by the theoryof similitude and dimensional analysis affordthe designer of experiments a unifying toolwhich will ordinarily reduce the number ofvariables in the problem and systematize therequirements of the simulation. These methods,however, do not remove the requirements forgood judgment, since there is no one path tofollow in their application to a given problem.An extensive modern literature on similitudeand dimensional analysis is available. References 5.1 through 5.3 are examples of generaltexts, while references 5.4 through 5.6 are examples of textbooks containing treatmentsoriented toward fluid dynamics (this .list is byno means exhaustive).5.9 REVIEW OF DIMENSIONAL ANALYSIS ASCUSTOMARILY APPLIED TO SIMULATIONAny physical measurement has two generalcharacteristics : qualitative and quantitative.The qualitative characteristics serve to identifythe class of quantities with which the measurement is to be associated and are called thedimensions of the quantity measured.
Thequantitative part of the measurement involvesa number and an arbitrary standard of comparison called a unit (the measurement; "3 in."has dimensions of length and units of inches).Insofar as Newtonian mechanics is concerned,the four dimensional categories of Force,F; Mass, M; Length, L; and Time, T,are related through the second law of motionor dimensionally:(where the symbol ( 5 ) denotes dimensionalbut not necessarily numerical equivalence) so145Preceding page blank &146THE DYNAMIC BEHAVIOR OF LIQUIDSt,hat any one of the four may be defined interms of the other three.
I t is convenient andtraditional to regard three of these four categories as independent, basic dimensions. Inmany analyses, force, length, and time areconsidered basic-inas many others, mass,length, and time are taken. The choice is amatter of convenience, though the M-L-Tsystem has been most often used in sloshingsimulation analyses. For problems involvingheat or electricity, additional basic quantities(temperature, e; quantity of heat, H; charge,Q; etc.) are often considered as basic dimensions. With conventional measuring techniques, the dimensions of any mechanicalquantity may be expressed as a product ofintegral powers of the basic dimensions.The methods of dimensional analysis arebuilt on the principle of dimensional homogeneity (an equation expressing a physicalrelationship between quantities must bedimensionally homogeneous).
The formalmethods were probably started by Lord Rayleigh (ref. 5.7) and improved upon by Buckingham (ref. 5.8) with a broad generalizationknown as the *-theorem. The *-theorem,in general terms, states that the -number ofdimensionless and independent quantities required to express a relationship among thevariables in any phenomenon is equal to thenumber of quantities involved, minus the number of independent dimensions in which thosequantities may be measured. These dimensionless quantities are comnlonly called "Tterms," and the only restriction on them isthat they be dimensionless and independent.There are a variety of methods in commonuse for deriving valid sets of *-terms from agiven list of variables (one or more dependent)and parameters. A commonly accepted formalprocedure which insures independence is as follows, given a list of n variables (al, a*, .
. . a,)containing m fundamental dimensions.The conceptual functional relationship of then variables may be written:I t is desired to form a function of *-termswhich is an equally valid representation of thephenomenonEach *-term must have the formwhere the x k are to be determined. Eachdimensionallyis(where the Djdenote the m fundamental dimensions, L, T, M , e, e t ~ . ) . Thus the general*-term is dimensionallyIf T is to be dimensionless, the combined exponent on each of the dimensions D jmust bezero. Thus, for j= 1, 2, .
. . m,Since equation (5.8) represents a system of mlinear algebraic equations in n unknowns (thezk), it is possible, generally, to solve thissystem of equations for the first m values of s tin terms of linear combinations of the remainingn-m unknown exponents. This general solution may be written for k = l , 2, . . . m;n5k=C cvgvp=m+l(5.9)\vl~erethe c p k are functions of the ajk and areccllculsble in general, if the determinant of t,hecoeffi~ientsa,~(j=1,2. . . m;k=1,2 . .
. m)is nonzero. If the original list of variablescontains each of the basic dimensions in moret l ~ a none \r&riable, and if the basic dimensionsdo not always occur in the same combinationin the dimensional representation of the variables, it will be possible to make a choice of mof the \1nknown exponents x k such that theSIMULATION AND EXPERIMENTAL TECHNIQUESsolution, equation (5.9), is possible. The x, inequation (5.9) are arbitrary. Thus, there arean unlimited number of solutions for the set ofexponents, xk, (li= 1, 2 . . .
m), since the set ofexponents, x,, ( p = m + l , . . . n), may bechosen in a n unlimited number of ways. Thus,it is possible to form a solution by settingX,+I equal to unity and all the other x, tozero. This process can be repeated by settingeach of the x, equal to unity in turn. Theresult is n-m sets of solutions for the esponentsxk, (k = 1, 2, . .
. m), as follows:The theory of linear equations states that thereare n -m linearly independent solutions toequation (5.9). The n-m sets of solutionsshown in equations (5.10) constitute n-m suchlinearly independent sets of solutions, probablythe simplest set. If a member of this set(eqs. (5.10)) is replaced by itself multiplied byany number other than zero, the resulting setremains a linearly independent one. Similarlyany member of the set of solutions (eqs. (5.10))may be replaced b it.s sum with any or all ofthe other members of the set with the sameresult.T h e general *-term may be written by(5.5)substituting equation (5.9) into eql~at~ionX [afnl agnz .. .
a:sra,,pn.ifor [i=l-,2,...(n-m)](5.11)I n this expression, t,he subscript i has beenadded to denote n-m particular choices of thearbitrary exponents xk. Equations (5.10) showthat one permissible choice for t'hese exponentsyields the n-m bracketed terms in equation(5.11) for the n-m independent n-terms.Other choices for the arbitrary esponents arejust as valid so long as the set of xk chosen will147result in n-?n linearly independent solutionsof equation (5.9).I t is evident from the form of equation (5.11)and the conditions on the linearly independentsolutions of equation (5.9) that so long asreplacenlent operations are carried out oneat a time:(1) Any member of a valid set of n-termsmay be replaced by itself raised to any (nonzero) power; and(2) Any member of a valid set of n-termsmay be replaced by itself raised to any nonzeropower and multiplied by any or all of the othermembers of the set, each raised to any power.I n general, while two analysts performing ttsystematic dimensional analysis of the sameset of variables may come up with valid setsof n-terms having different form, they musteach have the same number of terms, and itwill be possible to transform the results of thefirst analyst into those of the second; in otherwords, the formal methods of dimensionalanalysis imply that alternative solutions for avalid set of n-terms are not independent, andhence no one of them can be expected to contain any essential information not common tothe others.There are some obvious restrictions on thenature of the set of m variables chosen to berepeating in equation (5.11).
The dimensionsof these variables must include all the, m,fundamental dimensions, Dj, a t least once.Also, no two of these repeating variables canhave the same dimensional form nor should itbe possible to form a dimensionless group fromthese variables alone (these are the primaryconditions for a solution for the cpkcoefficients).An additional consideration is that if onlyone variable, a k , has the basic dimension D jin its dimensional form, then the conceptualfunctional relationship between the variablesand parameters ieq.
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