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## CH.7 Similitude, Dimensional Analysis, and Modeling

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**CH.7 Similitude, Dimensional Analysis, and Modeling**(相似律, 次元해석, 모델링) 7.1 Dimensional Analysis 1. Case Study; pressure drop per unit length in steady flow of an incompressible Newtonian fluid through a long, smooth-walled, horizontal, circular pipe ; pressure drop per unit length 2. Illustrative plots showing how the pressure drop in a pipe may be affected by several different factors ; see p398 Fig. 7.1 3. An illustrative plot of pressure drop data using dimensionless parameters (see p399 Fig. 7.2) ; Thus, instead of having to work with five variables, we now have only two.**4. 2 Nondimensional Group**5.Dimensional Analysis # Not only have we reduced the numbers of variables from five to two, but the new groups are dimensionless combinations of variables, which means that the results presented in the form of Fig. 7.2 will be independent of the system of units. # This type of analysis is called dimensional analysis, and the basis for its application to a wide variety of problems is found in the Buckingham pi theorem described in the following section.**7.2 Buckingham Pi Theorem**• - Theorem • If an equation involving variables is dimensionally homogeneous, it can be reduced to a relationship among • independent dimensionless products, where is the minimum number of reference dimensions required to describe the variables • Functional Relationship**7.3 Determination of Pi Terms**Method of Repeating Variables**Example 1 :*** total variables ; *dependent variable ; 1 ; *independent variables k-1=4 ; Step 1 ; List all the variables that are involved in the problem. Step 2 ; Express each of the variables in terms of basic dimensions Note! ; * For system, a dimension should occur at least one more time or not at all on both sides. In the above example, appears on each side of at least one more time.**So is ill-stated since pressure**involves the dimensions of mass(or force) and do not contain such a dimension. Step 3 ; Determine the required number of pi terms k-r=5-3=2 where r=3=number of reference(=basic) dimension. Step 4 ; Select a number of repeating variables, where the number required is equal to the number of reference dimensions. * # of repeating variables *3 repeating variables include all of basic dimensions, and are selected from the independent variables but they must not form a -parameter by themselves. ( 의 차원을 보면 가 각각 한번, 이 3번 나타나므로 각 차원이 2번 이상 나타나거나 아니면 아예 나타나지 않아야 한다는 pi - parameter 가 되기 위한 조건에 위배되므로 성립) * 2 nonrepeating variables ; dependent variable and**Step ５; Form a pi term by multiplying one of the**non-repeating variables by the product of the repeating variables, each raised to an exponent that will make the combination dimensionless. - Essentially each pi term will be of the form where is one of the non-repeating variables; and are the repeating variables ; and the exponents and and are determined so that the combination is dimensionless. - Typically, we would start with the dependent variable : for M ; 1+c=o, L ; -2+a+b-3c=o, T ; -2-b=o so a=1, b=-2, c=-1 for F ; 1+c=o, L ; -3+a+b-4c=o, T ; -b+2c=o so a=1, b=-2, c=-1**Step 6 ; Repeat Step 5 for each of the remaining**non-repeating variables. - resulting set of pi terms = k-r-1=5-3-1=1 - from this ; a=-1, b=-1, c=-1 Step 7 ; Check all the resulting pi terms to make sure they are dimensionless.**Step 8 ; Express the final form as a relationship among the**pi terms, and think about what it means. Typically the final form can be written as where would contain the dependent variable in the numerator. So**Example 2 ; (see Potter p232 Ex. 6.1)**The drag force on a cylinder of diameter and length is to be studied. What functional form relates the dimensionless variable? 1] Write the functional form of the dependent variable depending on the (k-1) independent variables. from experience ; * dependent variable ;*** Independent variable ;**• Note ! ; A dimension should occur at least one more time or not at all on both sides. • In this case ; • ; occurs 3 times, ; occurs 3 times, ; occurs 6 times • So appears on each side of at least • one more time. • 2] Identify r repeating variables, variable that will be combined with each remaining variable to form the pi-parameters. The repeating variables selected from the independent variables must include all of basic dimensions, but they must not form a pi-parameter by themselves.**1> Dimension of Variables**: Basic Dimension = 2> Identify r repeating variables ; * 3 repeating variables include all of basic dimensions, and are selected from the independent variables but they must not form a pi-parameter by themselves.( 의 차원을 보면 이 각각 한번 이 3번 출현하므로 각 차원이 2번 이상 출현하던가 아예 나타나지 않아야 한다는 pi-parameter가 되기 위한 조건에 위배되므로 성립) * can not be r repeating variables because they do not involve all of basic dimensions. 3] Form the pi-parameters by combining the repeating variables with each of the remaining variables. 1> each of the remaining variables ;**Note! ; It is convenient to set the value of exponent of**(=nonrepeating variables) as one. 2> number of pi-parameters(=dimensionless group) =k–r=6-3=3 3> first pi-parameter : from 4> second pi-parameter ;**From**5> third pi-parameter ;**4] Write the functional form of the (k - r) dimensionless**pi-parameters ; Final functional relationship is ; i.e.**7.6 Common Dimensionless Groups in Fluid Mechanics**Froude Number * derivation ; p414 * The Froude number is an index of the ratio of the force due to the acceleration of a fluid particle to the force due to gravity(=weight) * It is a measure of, or index of, the relative importance of inertial forces acting on fluid particles to the weight of the particle. Note that the Froude number is not really equal to this force ratio, but is simply some type of average measure of the influence of these two forces.*** In a problem in which gravity (or weight) is not**important, the Froude number would not appear as an important pi-term. • * It will generally be important involving flows with free surfaces since gravity principally affects this type of flow ; flow of water around ships, flow through rivers or open conduits • 2. Reynolds number : • * most famous dimensionless parameter in fluid mechanics • * If the Reynolds number is very small , this is an indication that the viscous forces are dominant in the problem, and it may be possible to neglect the inertial effects; that is, the density of the fluid will not be an important variable. : creeping flows • * Conversely, for large Reynolds number flows, viscous effects are small relative to inertial effects and for these cases it may be possible to neglect the effect of viscosity and consider the problem as one involving a “nonviscous” fluid.**3. Euler Number**• * Some form of the Euler number would normally be used in problems in which pressure or the pressure difference between two points is an important variable. • * • 4. Cauchy Number and Mach Number • * When the Mach number is relatively small(say, less than 0.3), the inertial forces induced by the fluid motion are not sufficiently large to cause a significant change in the fluid density, and in this case the compressibility of the fluid can be neglected.**5. Strouhal Number*** a dimensionless parameter that is likely to be important in unsteady, oscillating flow problems in which the frequency of the oscillation is * It represents a measure of the ratio of inertial forces due to the unsteadiness of the flow (local acceleration) to the inertial forces due to changes in velocity from point to point in the flow field (convective acceleration). * This type of unsteady flow may develop when a fluid flows past a solid body placed in the moving stream. * e.g. a periodic flow in the Karman vortex trail**6. Weber Number*** may be important in problems in which there is an interface between two fluids * In this situation the surface tension may play an important role in the phenomenon of interest.**Leonhard Euler**Born: 15 April 1707 in Basel, SwitzerlandDied: 18 Sept 1783 in St Petersburg, Russia**Euler was one the leading mathematicians of the 18th**century. Although the majority of his work was in pure mathematics, he contributed to other disciplines, such as astronomy and physics, as well. In his lifetime he published more than 500 books and papers, and another 400 were published posthumously(사후에). Euler was born in Basel, Switzerland on April 15, 1707. Although his father, a pastor, was a gifted amateur mathematician he wanted his son to succeed him in the village church. Despite his love for mathematics, Euler entered the University of Basel to study theology(신학). As a student, he attracted the attention of the Swiss mathematician, Johann Bernoulli. Bernoulli was able to convince the elder Euler to allow his son to drop his theological training and instead study mathematics. Soon after receiving his degree he was invited by the Empress of Russia in 1727 to be a professor of physics, and later mathematics, at the Academy of Sciences in St. Petersburg. He left for Berlin in 1741 and became professor of mathematics at the Berlin Academy of Sciences. He later returned to St. Petersburg at the urging of Catherine the Great in 1766. He lived there until he suffered a stroke and died in 1767. In his early 20's Euler lost his vision in one eye due to an illness. Later in life he lost the other eye, but despite**this he was quite productive in his field until his death.**Euler's contributions to mathematics cover a wide range, including analysis and the theory of numbers. He also investigated many topics in geometry.**Osborne Reynolds**Born: 23 Aug 1842 in Belfast, IrelandDied: 21 Feb 1912 in Watchet, Somerset, England**Osborne was born in Belfast where his father was Principal**of the Collegiate School there but began his schooling at Dedham when his father was headmaster of the school in that Essex town. After that he received private tutoring to complete his secondary education. He did not go straight to university after his secondary education, however, but rather he took an apprenticeship(도제기간) with the engineering firm of Edward Hayes in 1861. Reynolds, after gaining experience in the engineering firm, studied mathematics at Cambridge, graduating in 1867. As an undergraduate Reynolds had attended some of the same classes as Rayleigh who was one year ahead of him. As his father had before him, Reynolds was elected to a scholarship at Queens' College. He again took up a post with an engineering firm, this time the civil engineers John Lawson of London, spending a year as a practicing civil engineer. In 1868 Reynolds became the first professor of engineering in Manchester (and the second in England). Reynolds held this post until he retired in 1905. His early work was on magnetism and electricity but he soon concentrated on hydraulics and hydrodynamics. He also worked on electromagnetic properties of the sun and of comets(혜성), and considered tidal motions in rivers. After 1873 Reynolds concentrated mainly on fluid dynamics and it was in this area that his contributions were of world leading importance.**... Despite his intense interest in education, he was not**a great lecturer. His lectures were difficult to follow, and he frequently wandered among topics with little or no connection. Lamb, who knew Reynolds well both as a man and as a fellow worker in fluid dynamics, wrote:- The character of Reynolds was like his writings, strongly individual. He was conscious of the value of his work, but was content to leave it to the mature judgment of the scientific world. For advertisement he had no taste, and undue pretension on the part of others only elicited a tolerant smile. To his pupils he was most generous in the opportunities for valuable work which he put in their way, and in the share of cooperation**Augustin Louis Cauchy**Born: 21 Aug 1789 in Paris, FranceDied: 23 May 1857 in Sceaux (near Paris), France**French mathematician, b. at Paris, 21 August, 1789; d. at**Sceaux, 23 May, 1857. He owed his early training to his father, a man of much learning and literary taste, and, at the suggestion of La Grange, who early detected his talents and took a lively interest in him, he received a good classical education at the Ecole Centrale du Panthéon in Paris. In 1805 he entered the Ecole Polytechnique, where he distinguished himself in mathematics. Two years later he entered the Ecole des Ponts et Chaussées and, after a brilliant course of study, he was appointed one of the engineers in charge of the extensive public works inaugurated by Napoleon at Cherbourg. While here he devoted his leisure moments to mathematics. Several important memoirs from his pen, among them those relating to the theory of polyhedra(다면체), symmetrical functions, and particularly his proof of a theorem of Fermat which had baffled (좌절)mathematicians like Gauss and Euler, made him known to the scientific world and won him admittance into the Academy of Sciences. At about the same time the Grand Prix offered by the Academy was bestowed on him for his essays on the propagation of waves. After a sojourn(체재) of three years at Cherbourg his health began to fail, and he resigned his post to begin at the age of twenty-two his career of professor at the Ecole Polytechnique. In 1818 he married Mlle. de Bure, who, with two daughters, survived him.**Cauchy was a stanch adherent of the Bourbons and after**the Revolution of 1830 followed Charles X into exile. After a brief stay at Turin, where he occupied the chair of mathematical physics created for him at the university, he was invited to become one of the tutors of the young Duc de Bordeaux, grandson of Charles, at Prague. The old monarch conferred the title of baron upon him in recognition of his services. He returned to France in 1838, and was proposed by the Academy for a vacant chair at the Collège de France. His conscientious refusal to take the requisite oath on account of his devotion to the prince prevented his appointment. His nomination to the Bureau des Longitudes was declared void for the same reason. After the Revolution of 1848, however, he received a professorship at the Sorbonne. Upon the establishment of the Second Empire the oath was reinstated, but an exception was made by Napoleon III in the cases of Cauchy and Arago, and he was thus free to continue his lectures. He spent the last years of his life at Sceaux, outside of Paris, devoting himself to his mathematical researches until the end.**Froude, William (1810-1879)**English engineer and hydrodynamicist who first formulated reliable laws for the resistance that water offers to ships and for predicting their stability. He also invented the hydraulic dynameter (1877) for measuring the output of high-power engines. These achievements were fundamental to marine development. Froude was born in Devon and educated at Oxford. He remained for a time at Oxford, working on water resistance and the propulsion of ships. From 1859 he carried out tank-testing experiments at his home, first in Paignton and then in Torquay. In 1838 Froude assisted Isambard Kingdom Brunel on the building of the Bristol and Exeter Railway. Brunel later consulted him on the behaviour of the Great Eastern at sea and, on his recommendation, the ship was fitted with bilge keels. Beginning in 1867, Froude towed models in pairs, balancing one hull shape against the other. Realizing that the frictional resistance and the wave-making resistance follow different laws, he also towed submerged planks with different surface roughness. He was able to establish a formula which would predict the frictional resistance of a**hull with accuracy, and formulated Froude's law of**comparison. This stated that the wave-making resistance of similar-shaped models varies as the cube of their dimensions if their speeds are as the square root of their dimensions. With these two analytical results, Froude had found a reliable means of estimating the power required to drive a hull at a given speed. Froude also carried out model experiments and theoretical work on the rolling stability of ships. His general deductions are still the standard exposition of the rolling and oscillation of ships.**Ernst Mach (1838-1916) spent 28 years in Prague between**1867 and 1895. He contributed to the development of several parts of physics including optics, acoustics and mechanics. Ernst Mach was born on 18 February 1838 in Chrlice (nowadys a part of the Moravian metropolis Brno) in a German family whose name, however, betrays Czech ancestry. After completing his studies at Kromeriz Gymnasium, he enrolled for study at the University of Vienna where he studied mathematics, physics and philosophy. His early physical works were devoted to electric discharge and induction. Between 1860 and 1862 he studied in depth the Doppler Effect by optical and acoustic experiments. Already this work proved Mach's competence as a brilliant experimenter and a designer of measuring devices striving for maximum precision. In 1864 Mach became Professor of Mathematics in Graz, in 1866 he was named Professor of Physics. During this period Mach was interested also in physiology of sensory perception. In 1867 Mach became Professor of Experimental Physics at the Prague Charles University (called the Karl - Ferdinand at that time). His lectures, accompanied by well-premeditated experiments, soon gained a reputation for their excellence and pedagogical acumen. In 1879/80 and 1883/84 Mach was elected Rector of the University. However, during his**second Rectorship he resigned since he disapproved of the**administrative procedure which was applied during the statutory division of Prague University into Czech and German parts beginning from the winter term 1882/83. Mach subsequently worked and lectured in the German part of the University (he was the head of the Institute of Physics) but his courses were attended by many Czech students. Mach's scientific interests during his Prague period covered a wide cross-section of physics. The list of his works written in Prague features about 90 publications including 6 books. Naturally, he continued his previous investigations. In Prague he published papers on the physiology of sensory organs. Some of the experiments Mach conducted on himself. Most of his studies in the field of experimental physics were devoted to acoustics and physical optics. They dealt with interference, diffraction, polarization and refraction of light in different media under external influences. At the same time, Mach resumed his study of the Doppler effect. He developed a number of excellent measuring instruments which were used in research and for instruction in universities. He also invented new experimental methods in stroboscopy and photography.**These studies were soon followed by his important**explorations in the field of supersonic velocity. Mach's paper on this subject was published in 1877 and correctly describes the sound effects observed during the supersonic motion of a projectile. Mach deduced and experimentally confirmed the existence of a shock acoustic wave which has the form of a cone with the projectile at the apex. The ratio of the speed v/c is now called the Mach number. It plays a crucial role in aerodynamics and hydrodynamics.**7.8 Modeling and Similitude**• - Definition of Engineering Model(모형), Prototype(원형) • * A model is a representation of a physical system that may be used to predict the behavior of the system(=prototype) in some desired respect. • * Models resemble the prototype but are generally of a different size, may involve different fluids, and often operate under different conditions(pressures, velocities, etc.). • prototype ; The physical system for which the predictions are to be made is called the prototype. • - It is imperative that the model be properly designed and tested and that the results be interpreted correctly.**Karman, Theodor von , 1881–1963, American aeronautical**engineer, born. Hungary, grad. Royal Technical Univ., Budapest (1902), and Univ. of Göttingen, Germany (Ph.D., 1908). From 1909 to 1912 he served as director of the aeronautical institute at the Univ. of Aachen. He came to the United States in 1930, was naturalized in 1936, and was on the staff of the California Institute of Technology from 1930 to 1949. He made many contributions to the field of aerodynamics and is known especially for his mathematical formulas called the von Karman theory of vortex streets. These formulas are used in the calculation of the resistance by air to objects (e.g., aircraft, rockets) moving through it. His writings include Aerodynamics (1954) and his autobiography, Wind and Beyond (with Lee Edson, publ. posthumously, 1967).**7.8.1 Theory of Models**Description of Any Given Problem by a Set of Pi terms 2. Model Design Condition(=Similarity Requirements or Modeling Laws) and Prediction Equation - The pi terms can be developed so that contains the variable that is to be predicted from observations made on the model. Therefore, if the model is designed and operated under the following conditions, - then with the presumption that the form of is the same for model and prototype, it follows that