布莱休斯函数研究

Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. SIAM REVIEW c© 2008 Society for Industrial and Applied Mathematics Vol. 50, No. 4, pp. 791–804 The Blasius Function: Computations Before Computers, the Value of Tricks, Undergraduate Projects, and Open Research Problems∗ John P. Boyd† Abstract. The Blasius flow is the idealized flow of a viscous fluid past an infinitesimally thick, semi- infinite flat plate. The Blasius function is the solution to 2fxxx + ffxx = 0 on x ∈ [0,∞] subject to f(0) = fx(0) = 0, fx(∞) = 1. We use this famous problem to illustrate several themes. First, although the flow solves a nonlinear partial differential equation (PDE), Toepfer successfully computed highly accurate numerical solutions in 1912. His secret was to combine a Runge–Kutta method for integrating an ordinary differential equation (ODE) initial value problem with some symmetry principles and similarity reductions, which collapse the PDE system to the ODE shown above. This shows that PDE numerical studies were possible even in the precomputer age. The truth, both a hundred years ago and now, is that mathematical theorems and insights are an arithmurgist’s best friend, and they can vastly reduce the computational burden. Second, we show that special tricks, applicable only to a given problem, can be as useful as the broad, general methods that are the fabric of most applied mathematics courses: the importance of “particularity.” In spite of these triumphs, many properties of the Blasius function f(x) are unknown. We give a list of interesting projects for undergraduates and another list of challenging issues for the research mathematician. Key words. boundary layer, fluid mechanics, Blasius AMS subject classifications. 65-01, 76-01, 76D10 DOI. 10.1137/070681594 Thinking is the cheapest and one of the most effective long-range weapons. –Field Marshal Sir William Slim, 1st Viscount Slim (1891–1970) 1. Introduction. Slim’s maxim applies as well to science as to war. The Blasius problem of hydrodynamics is a good illustration of how cunning can triumph over brute force. The Blasius flow of a steady fluid current past a thin plate is the solution of the partial differential equation (PDE) (1.1) ψY Y Y + ψX ψY Y − ψY ψXY = 0, X ∈ [−∞,∞], Y ∈ [0,∞], where subscripts with respect to a coordinate denote differentiation with respect to the ∗Received by the editors February 1, 2007; accepted for publication (in revised form) February 7, 2008; published electronically November 5, 2008. This work was supported by NSF grants OCE 0451951 and ATM 0723440. http://www.siam.org/journals/sirev/50-4/68159.html †Department of Atmospheric, Oceanic and Space Science, University of Michigan, 2455 Hayward Avenue, Ann Arbor, MI 48109 (jpboyd@engin.umich.edu, http://www.engin.umich.edu/∼jpboyd/). 791 Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. 792 JOHN P. BOYD X Y Undisturbed Uniform Flow Boundary Layer Fig. 1.1 The Blasius flow is the result of the interaction of a current that is spatially uniform for large negative X (left part of diagram) with a solid plate (thin shaded rectangle), which is idealized as being infinitely thin and extending infinitely far to the right as X → ∞. Because all fluid flows must be zero at a solid boundary, the velocity must slow rapidly to zero in a “boundary layer,” which thickens as X → ∞. The region of velocity change (“shear”) is called a “boundary layer” because in fluids of low viscosity, such as air and water, the shear layer is very thin. coordinate (three subscripts for a third derivative, and so on), subject to the boundary conditions (1.2) ψ(X, 0) = 0, X ∈ [−∞, 0], ψY (X, 0) = 0, X ∈ [0,∞], ψY (X,∞) = 1, where ψ(X,Y ) is the so-called streamfunction; the fluid velocities are just the spatial derivatives of ψ. Figure 1.1 illustrates the flow schematically. The fluid mechanics background is given in all graduate and most undergraduate texts in hydrodynamics. Although the geometry is idealized, all flows past a solid body have thin “bound- ary layers” similar to the Blasius flow. Air rushing past a bird or an airplane, ocean currents streaming past an undersea mountain, a brook babbling through rapids made by a boulder and fallen trees, even the blood and breath flowing through our own bodies—all have boundary layers. The Blasius problem is as fundamental to fluid mechanics as the tangent function to trigonometry. The Blasius problem has developed a vast bibliography [10]. Even though the problem is almost a century old, recent papers that employ the Blasius problem as an example include [2, 1, 5, 6, 11, 15, 16, 21, 18, 17, 23, 25, 26, 27, 28, 29, 30, 32, 33, 34, 36]. In this age of fast workstations, it is an almost irresistible temptation to blindly apply a two-dimensional finite difference or finite element method and then, a few billion floating point operations later, make a gaudy full-color contour plot of the answer. Voila´! Unfortunately, numbers and pictures are meaningless by themselves. A supercom- puter is like a ten-foot slide rule: it adds a little more accuracy to the results of less ambitious hardware, but it is still not a substitute for thinking. For all computational problems, intricate contour lines and isosurface plots advance science no more than a toddler’s fingerpainting unless guided by a physicist’s intuition and ability to plan, supervise, and analyze. For the Blasius problem, the additional payoffs of thinking (or rethinking) are not merely an enormous reduction in the computational burden, but also a drastic simplification of the conceptual burden: it is easier to understand a function of one variable than a function of two. Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. THE BLASIUS FUNCTION 793 2. Symmetry Reductions. 2.1. PDE to ODE. Blasius himself noted in his 1908 paper that if ψ(X,Y ) is a solution, then so is (2.1) Ψ(X,Y ) ≡ c ψ(c2X, cY ), where c is an arbitrary constant. In other words, this implies that the problem has a continuous group invariance. The streamfunction is not a function of X and Y separately, but rather must be a univariate function of the “similarity” variable (2.2) x ≡ Y√ X . Any other dependence on the coordinates would disrupt the group invariance. There is some flexibility in the sense that one could replace x by x2 or any other smooth function of x as the similarity variable, but the most convenient choice, and the historical choice, is (2.3) ψ(X,Y ) = √ X f(Y/ √ X) for some univariate function f . Thus, Blasius showed that the PDE (1.1) could be converted to the ODE (2.4) 2fxxx + ffxx = 0 subject to the boundary conditions (2.5) f(0) = fx(0) = 0, fx(∞) = 1. Figure 2.1 illustrates the Blasius function and its derivatives. Blasius himself [7] derived both power series and asymptotic expansions and patched them together at finite x to obtain an approximation which agrees quite satisfactorily with later treatments. In particular, he computed the value of the sec- ond derivative at the origin to about one part in a thousand. Note that, in some problems (but not the Blasius problem), “symmetry-breaking” solutions are possible. Also, some boundary layers and other steady solutions (but not the Blasius flow) are unstable. These possible complications, which are com- mon throughout physics, reiterate how important it is for physical thinking to guide mathematical problem-posing. 0 2 4 6 8 0 2 4 6 0 2 4 6 8 0 0.2 0.4 0.6 0.8 0 2 4 6 8 0.1 0.2 0.3 f df/dx d2f/dx2 x x x Fig. 2.1 A plot of the Blasius function (left) and its first two derivatives (middle and right). Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. 794 JOHN P. BOYD 2.2. Conversion of the Boundary Value Problem to an Initial Value Prob- lem. Boundary value problems must be solved at all points simultaneously (a “jury” problem), whereas an initial value problem can be solved by a stepwise procedure (a “marching” problem). In this sense, initial value problems are easier. The Blasius ODE could be converted into an initial value problem and integrated by marching from x = 0 to large x if only (2.6) κ = fxx(0) were known. Karl Toepfer [35] realized that knowledge of κ is in fact unnecessary. The reason is that there is a second group invariance such that if g(x) denotes the solution to the Blasius equation with g(0) = 0, gx(0) = 0, and its second derivative is arbitrarily set equal to one, then the solution with fxx(0) = K is (2.7) f(x;K) = K1/3 g(K1/3x). It therefore suffices to compute g(x) and then rescale both the horizontal and vertical axes of the graph of g(x) so that the rescaled function has the desired asymptotic behavior at large x, namely, fx(∞) = 1. The true value of the second derivative at the origin is then (2.8) κ = lim x→∞ gx(x) −3/2. With Toepfer’s trick, it is only necessary to solve the differential equation as an initial value problem once. At two large but finite xj , ordered so that xj > xj−1, compute κj ≡ (1/gx(xj))3/2. If the κj closely agree, κ is approximately equal to the common value of the κj ; if they are far apart, march to still larger x and try again. Using a Runge–Kutta method with a grid step of 1/10 and the endpoints of x1 = 4 and x2 = 6, Toepfer was thus able to determine κ correctly to about one part in 110,000! Freshman physics books are populated with analytical solutions, but none has ever been found for the Blasius equation. The first general-purpose electronic computer, ENIAC, was more than a third of a century in the future when Toepfer did his work. Nevertheless, by exploiting group invariances, he reduced the Blasius problem to perhaps a couple of hundred multiplications and additions. Though he does not record his paper-and-pencil computing time, it was likely only an afternoon. 3. Lessons from Symmetry Reductions and the Numerical Solution. 3.1. When Computers Were Human. In conducting extensive arithmetical operations, it would be natural to avail oneself of the skill of professional [human] computers. But unfortu- nately the trained computer, who is also a mathematician, is rare. I have thus found myself compelled to forego the advantage of the rapidity and accuracy of the computer, for the higher qualities of mathematical knowl- edge and judgment. –Sir George C. Darwin (1845–1912) [14, p. 101] The Blasius–Toepfer numerical work was only a tiny part of the vast computations performed before the advent of electronic computers. Much of the number-crunching was done by full-time human calculators known as “computers.” Grier [20], Croarken [12], and others [3, 13] have written very readable accounts of the heroic age of number- crunching. A substantial portion of the “computers” were female—none of this “girls can’t do math” nonsense in the eighteenth or nineteenth century. Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. THE BLASIUS FUNCTION 795 Weyl [39, p. 385] wrote that the Blasius problem “was the first boundary-layer problem to be numerically integrated . . . [in] 1907.” However, the history of numerical solutions to differential equations is much older and richer. More than two decades before Blasius, John Couch Adams devised what are now called the Adams–Bashforth and Adams–Moulton methods for numerically integrating an initial value problem [4]. Toepfer numerically integrated the Blasius problem using a Runge–Kutta method published by Runge in 1895 and greatly refined by Kutta in 1901. Aspray notes in his preface to Computing Before Computers [3], “Wherever we turn we hear about the ‘Computer Revolution’ and our ‘Information Age’. . . . With all of this attention to the computer we tend to forget that computing has a rich history that extends back beyond 1945.” Similarly, Gear and Skeel [19] note that the post–World War II development of electronic computers “has, of course, affected the algorithms used, but this has resulted in surprisingly few innovations in numerical techniques.” What they mean is that although numerical algorithms have been greatly improved and advanced since the first electronic computers were built in the late 1940s, the building blocks—Runge–Kutta, Lagrangian interpolation, finite differences, etc.— were all created in the precomputer age. For example, Lanczos published his great paper, which is the origin of both Chebyshev pseudospectral methods and the tau method, in 1938 [24]. Rosenhead’s point vortex paper was published even earlier [31]. George Boole’s book [8] on finite differences first appeared in 1860! Howarth’s 1938 article, which contains an extensive table of the Blasius f(x) and its first two derivatives, contains the acknowledgment, “I wish to express my gratitude to the Air Ministry for providing me with a [human] computer to perform much of the mechanical labour” [22]. But his calculations by government-funded human computer had long antecedents. Sir George Darwin, legendary for his prodigious numerical cal- culations in celestial mechanics and the equilibria of self-gravitating stars and planets, independently reinvented some of Adams’ methods and used them to compute peri- odic orbits in 1897 [14]. He noted sadly that his previous human computer had died, but he found two new helpers at Cambridge and did much of the calculating him- self. Foreshadowing Howarth, he thanked a grant from the Royal Society for funding two-thirds of the cost of this monumental number-crunching. This brief history, though omitting many other examples, shows that computing has never been primarily about electronics, but always primarily about mathematics, algorithms, and organization, plus money for the he/she/it which is the “computer.” 3.2. Precomputing. Computing is a temptation that should be resisted as long as possible. –John P. Boyd in the first edition of his book Chebyshev and Fourier Spectral Methods (Springer-Verlag, 1989) Presoaking is a sound strategy for washing pots and pans with burnt or dried residues. In a similar way, successful computing is dependent on precomputing: iden- tifying symmetries and other mathematical principles that can greatly reduce the scope of the problem. Engineers have relied for generations on dimensional analysis and the Buckingham pi theorem, which assert that physics must be independent of the system of units. Thus, a problem with three dimensional parameters can be reduced to the computation of a function that depends on only one parameter or perhaps no parameters at all. Good algorithms are vital, but intelligent formulation of the numerical problem is equally important. Unfortunately, the vast expansion of numerical algorithms, soft- Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. 796 JOHN P. BOYD ware management, and parallel computing has rather crowded out of the curriculum the tools of twentieth century applied mathematics: singular perturbation theory, group theory, special functions, and transformations. The Blasius problem is but one of many examples where these fading tools reduced the computational burden by an order of magnitude and greatly simplified the description of results. A century of progress in floating point hardware has not reduced the need for problem-specific thinking, but rather merely increased our number-crunching ambitions. 4. Known Properties. It is important to make friends with the function. –Tai Tsun Wu, Gordon McKay Professor of Applied Physics, Harvard Before we can dive into the virtues of “particularity,” we need to “make friends” with the Blasius function. As archaeologists reconstruct a lost civilization one pottery shard at a time, so applied mathematics accumulates understanding through a slow accretion of individual properties. 4.1. Power Series. The power series begins (4.1) f(x) ≈ κ 2 x2 − κ 2 240 x5 + 11 161280 κ3x8 − 5 4257792 κ4x11 + · · · , where κ is the second derivative of the function at the origin, given to very high precision in Table 4.1. Only every third coefficient is different from zero. As proved in introductory college calculus, a power series converges only within the largest disk, centered on x = 0, which is free of singularities of f(x). Although it would take us too far afield to recapitulate the proof here, the Blasius function has a singularity on the negative real axis [10] at x = −S, where no closed form for S is known. The series converges for |x| < S, where S ≈ 5.69, and is given to high precision in the table. A closed form for the coefficients is not known. However, denoting the general series by f = ∑∞ j=0 aj x 3j+2, the coefficients can be computed from a1 = κ/2 plus the recurrence (4.2) am = − 1(3m− 1)(3m− 2)(3m− 3) m−1∑ j=1 (3j − 1) (3j − 2) aj am−j . The limitation of a finite radius of convergence can be overcome by constructing, from the power series, either Pade´ approximants or an Euler-accelerated series, which both apparently converge for all positive real x [10, 9]. 4.2. Asymptotic Approximations. For large positive x, (4.3) f(x) ∼ B + x+ exponentially decaying terms, x 1, and (4.4) fxx = Q exp {−(1/4)x(x+ 2B)} , Table 4.1 Blasius constants to high precision (for benchmarking). Symbol Definition Numerical value κ fxx(0) 0.33205733621519630 S power series radius of convergence 5.6900380545 B limx→∞(f(x)− x) −1.720787657520503 Q limx→∞ exp {[1/4]x(x+ 2B)} f(x) 0.233727621285063 Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. THE BLASIUS FUNCTION 797 -3 -2 -1 0 1 2 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 0.1 0.2 0.30.4 0.5 0.6 0.70.8 0.9 1 1 Re(y) [y=x/S] Equiconvergence contours: Eulerized series • • • • • • • Fig. 4.1 The Blasius function in the complex plane. The coordinate is scaled by S, the distance from the origin to the nearest singularity. The three thick rays are the symmetry axes in the complex plane. The black dots denote known singularities of the Blasius function. The contours are the “equiconvergence” contours of the Euler-accelerated power series; everywhere along the contour labeled “0.8,” the n+ 1th term is smaller than the nth term by a factor of 0.8. The Eulerized series appears to converge everywhere within the region bounded by the three dashed parabolas (including the entire positive real axis, which is the physical domain for the flow), but a rigorous proof is lacking. where Q ≈ 0.234 and B ≈ −1.72. (These constants are given to high precision in Table 4.1.) The linear polynomial B+x, which is the leading order asymptotic approximation for f(x), is an exact solution to the differential equation for all x, but fails to satisfy the boundary conditions. Physically, the Blasius velocity fx is constant at large distances from the plate, and this implies that f(x) should asymptote to a linear function of x far from the plate. Near the surface of the plate at x = 0, the streamfunction f curves away from the straight line to satisfy the boun