{"id":1313,"date":"2020-05-29T23:30:39","date_gmt":"2020-05-30T03:30:39","guid":{"rendered":"http:\/\/underthehood.blogwyrm.com\/?p=1313"},"modified":"2022-07-28T06:27:51","modified_gmt":"2022-07-28T10:27:51","slug":"prandtls-boundary-layer","status":"publish","type":"post","link":"https:\/\/underthehood.blogwyrm.com\/?p=1313","title":{"rendered":"Prandtl’s Boundary Layer"},"content":{"rendered":"

This month\u2019s column begins a new chapter in the ongoing study of fluid dynamics.\u00a0 Up to this point, the general equations of fluid motion have been derived from Newton\u2019s laws using the transformation between a fluid-following viewpoint (Lagrangian point-of-view) and a field concept (Eulerian point-of-view).\u00a0 General considerations of mass conservation and the principles of momentum and energy led to a system of non-linear partial differential equations.\u00a0 Ignoring viscosity allowed the equations to simplify to Euler\u2019s equations for an ideal fluid and several specific examples drove home some of the kinematics, dynamics, and methods of solution.<\/p>\n

Unfortunately, the situation changes quite dramatically when viscosity is brought back into consideration and the full Navier-Stokes equations (which are themselves something of a simplification) begin to stare back from the page.\u00a0 What is the general mode of attack, and how does one think about the solutions that result?\u00a0 The enormous difficulty of finding solutions becomes even more acute when two facts are driven home.\u00a0 First, it is extremely difficult to find solutions for the simpler set of equations for an ideal fluid, and the solutions shown to date (e.g. the Rankine Vortex<\/a>), complicated as they may seem, are special cases passed down from one generation to the next precisely because they had a solution.\u00a0 Second, general analytic solutions remain elusive nearly 200 years after Navier first derived these equations in 1822 despite millions of dollars offered to those who could derive such solutions.<\/p>\n

To set the stage for an exploration that will consume many columns to come, this installment will present some of the observations and follow many of the arguments gleaned from the article Ludwig Prandtl\u2019s Boundary Layer<\/em><\/a> by John D. Anderson Jr published in the December 2005 issue of Physics Today.<\/p>\n

According to Anderson, Ludwig Prandtl<\/p>\n

\"\"<\/a><\/p>\n

revolutionized fluid dynamics with a very simple to understand and yet highly non-intuitive concept of the boundary layer that exists in a fluid near a rigid surface.<\/p>\n

To understand the importance of Prandtl\u2019s result one has to start the story in the latter half of the 18th<\/sup> century, around the time Euler derived the ideal fluid equations bearing his name.\u00a0 The French mathematician Jean le Rond d\u2019Alembert derived a paradox<\/a> associated with ideal fluid flow.\u00a0 He proved that, for an incompressible fluid with no viscosity (i.e., an ideal fluid), the drag force on an object placed within this flow, moving with constant relative velocity with respect to the fluid, is identically zero, an obviously incorrect result.<\/p>\n

Acheson presents a mathematical derivation of d\u2019Alembert\u2019s paradox In Section 4.13 of his book Elementary Fluid Dynamics<\/em>.\u00a0 This derivation examines the force and momentum balance in the fluid and, using Bernoulli\u2019s streamline theorem<\/a>, concludes that, for the fluid to be moving at a uniform speed far upstream and far downstream from the body, the drag force on the body must vanish.\u00a0 This result contradicts everyday observations about the motion of solid objects in air or water.<\/p>\n

Clearly something was wrong with the theory of ideal fluid, but the specific root of the problem escaped identification for roughly 150 years until Prandtl introduced his boundary layer at a conference in 1904.\u00a0 The modern picture of the boundary layer is best understood in terms of the motion of air over a cambered wing as shown in the figure below (based on Figure 2 of Anderson).<\/p>\n

\"\"<\/a><\/p>\n

Away from the wing, the ideal fluid model works well in describing the streamlines.\u00a0 However, close to the wing\u2019s surface, viscous effects become dominant in a thin layer whose base attaches to the wing\u2019s surface.\u00a0 This no-slip condition demands that the fluid be at rest at the surface and that the fluid velocity rapidly rise from zero (where the layer attaches to the wing) to the \u2018free stream\u2019 velocity of the bulk inviscid flow outside the layer (shown in the inset).\u00a0 In this way, d\u2019Alembert\u2019s paradox is conceptually resolved.\u00a0 The drag force is non-zero, as observed, and the bulk flow is inviscid or ideal (depending on whether one allows the viscosity-free flow to be compressible or not), as observed, but the mathematical conditions by which d\u2019Alembert derived the paradox are not true everywhere \u2013 the thin boundary layer makes all the difference.<\/p>\n

Prandtl also predicted that, in certain cases, this layer can separate from the surface, which can cause additional drag.\u00a0 To quote Prandtl<\/p>\n

\u2026there ought to be a layer of fluid which, having been set in rotation by the friction on the wall, insinuates itself into the free fluid, transforming completely the motion of the latter, and therefore playing there the same part as the Helmholtz surfaces of discontinuity.<\/div>\n

Indeed, modern theory classifies the boundary layer behavior as falling into two broad classes.\u00a0 Laminar motion describes the flow when the layer stays fixed to the surface.\u00a0 Turbulent motion describes the flow when the layer separates and produces vortices in wake.\u00a0 Laminar flow produces less drag than turbulent flow but it is unstable and thus harder to produce or control.\u00a0 Turbulence changes the characteristics of the flow far from the surface causing additional drag.<\/p>\n

In addition to these two critical physical observations, Anderson points out that the concept of the boundary layer also enabled the first quantitative calculations of aerodynamic drag. \u00a0The reason for this is that the boundary layer equations exhibit an entirely different type of mathematical behaviors than the equations describing the bulk flow outside.\u00a0 Although both sets of equations are coupled, nonlinear, partial differential equations, the Navier-Stokes equations used to describe the bulk flow external to the layer are elliptic (at least when the flow is subsonic) while the simplification of these equations in the boundary layer are parabolic.\u00a0 This distinction allows the solution methods used to describe the bulk motion to \u2018decoupled\u2019 from the methods used to describe the drag and since the methods for solving parabolic equations are much easier to deal with than those for elliptic equations real progress can be made.<\/p>\n

What is particularly amazing about Prandtl\u2019s contribution is that the original presentation in 1904 lasted no longer than 10 minutes.\u00a0 I would say that the return on investment resulting from that rather short talk have been nothing short of phenomenal.<\/p>\n","protected":false},"excerpt":{"rendered":"

This month\u2019s column begins a new chapter in the ongoing study of fluid dynamics.\u00a0 Up to this point, the general equations of fluid motion have been derived from Newton\u2019s laws… Read more ><\/a><\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-1313","post","type-post","status-publish","format-standard","hentry","category-uncategorized"],"_links":{"self":[{"href":"https:\/\/underthehood.blogwyrm.com\/index.php?rest_route=\/wp\/v2\/posts\/1313","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/underthehood.blogwyrm.com\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/underthehood.blogwyrm.com\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/underthehood.blogwyrm.com\/index.php?rest_route=\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/underthehood.blogwyrm.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=1313"}],"version-history":[{"count":4,"href":"https:\/\/underthehood.blogwyrm.com\/index.php?rest_route=\/wp\/v2\/posts\/1313\/revisions"}],"predecessor-version":[{"id":1343,"href":"https:\/\/underthehood.blogwyrm.com\/index.php?rest_route=\/wp\/v2\/posts\/1313\/revisions\/1343"}],"wp:attachment":[{"href":"https:\/\/underthehood.blogwyrm.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1313"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/underthehood.blogwyrm.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=1313"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/underthehood.blogwyrm.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=1313"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}