Submitted - N-71.pdf

", "abstract": "In many fluid flows of practical importance in engineering, agriculture, and meteorology, the presence in the flow of foreign matter which is transported by the flow is of great importance. Erosion by flood waters is one important example, the pollution of masses of water or atmosphere is another. The importance of the transported matter arises in various ways; in some cases principal interest\ncenters on the quantity of an impurity removed by a flow, while in others a determination of the characteristics of the main flow itself depends to a large extent on the quantity and nature of solid matter\nentrained by the flow and carried along in it. The possibility of controlling or predicting such phenomena depends on a knowledge of the physical mechanisms of the processes which are involved. At the present time such knowledge is insufficient to cope with important\npractical problems.\n\nWhen the foreign matter consists of solid particles, as in problems of erosion and sedimentation, it is natural to seek an analogy with molecular processes by supposing that the history of each particle or group of particles is characterized by a randomness of the same type present in molecular kinetics. A diffusion theory is thereby obtained,\nthe practical value of which is, of course, determined solely by its ability to explain observed phenomena. Under some circumstances both qualitative and quantitative agreement is found to be good; in other important cases the agreement is less than satisfactory. Similar\nanalogies are used when one considers processes near boundaries at which foreign matter is entrained into and leaves the flow. One then draws analogy with the recognized theories of turbulent transfer of momentum, heat, and vorticity. These attempts likewise meet with\nvarying degrees of success.\n\nExperiments in sedimentation (Vanoni, 1946) have shown not only limits of the theories which have so far been applied, but they have also indicated roughly the dependence of observed discrepancies on parameters in some cases where the analogy theories fail. These circumstances\nmay facilitate attempts to obtain more refined theories. As a first step toward obtaining such refinements, the investigation described by the present report was conducted with a dual purpose : (1) to make an examination of the fundamental physical and mathematical features of\nclassical molecular diffusion theory, with particular regard to the complications which are believed to be at the base of observed discrepancies in analogy theories and (2) to survey the work which has been done in\nallied fields, especially meteorology, in which similar problems have been faced for many years by competent physicists and mathematicians, in order to see which refinements may be carried over to the theory of\nsedimentation and turbulent diffusion and to suggest new lines of attack which may be fruitful.\n\nAs the difficulties with the theory are largely due to great mathematical complications in any but the simplest problems, the discussion which follows places emphasis on mathematical techniques. In order to\nmake the presentation more readable to workers in hydraulics who do not make frequent use of modern advanced techniques of mathematical physics, the discussion is confined to a moderately elementary level.", "date": "2015-07-22", "date_type": "published", "publisher": "California Institute of Technology", "id_number": "CaltechAUTHORS:20150629-105950661", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20150629-105950661", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "Office of Naval Research (ONR)", "grant_number": "N6onr-24431" } ] }, "collection": "CaltechAUTHORS", "other_numbering_system": { "items": [ { "id": "89", "name": "Hydrodynamics Laboratory" } ] }, "local_group": { "items": [ { "id": "Hydrodynamics-Laboratory", "value": "Hydrodynamics Laboratory" } ] }, "primary_object": { "basename": "N-71.pdf", "url": "https://authors.library.caltech.edu/records/cdwz6-4kr89/files/N-71.pdf" }, "resource_type": "monograph", "pub_year": "2015", "author_list": "Michelson, Irving" }, { "id": "https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/zt6sp-jyw37", "eprint_id": 450, "eprint_status": "archive", "datestamp": "2023-08-21 22:26:13", "lastmod": "2023-10-13 21:49:14", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Marble-F-E", "name": { "family": "Marble", "given": "Frank E." } }, { "id": "Michelson-I", "name": { "family": "Michelson", "given": "Irving" } } ] }, "title": "Analytical Investigation of Some Three-Dimensional Flow Problems in Turbomachines", "ispublished": "unpub", "full_text_status": "public", "note": "Guggenheim Jet Propulsion Center Publication No. 10", "abstract": "One problem encountered in the theory of turbomachines is that of calculating the fluid velocity components when the inner and outer boundaries of the machine as well as the shape of or forces imparted by the blade row are given. The present paper discusses this problem under the restrictions that the fluid is inviscid and incompressible and that the blade rows consist of an infinite number of infinitely thin blades so that axially symmetric flow is assumed. \n\nIt is shown, in general, that the velocity components in a plane through the turbomachine axis may be expressed in terms of the angular momentum and the leading-edge blade force normal to the stream surfaces. The relation is a nonlinear differential equation to which analytic solutions may be obtained conveniently only after the introduction of linearizing assumptions. A quite accurate linearization is effected through assuming an approximate shape of the stream surfaces in certain nonlinear terms. \n\nThe complete linearized solution for the axial turbomachine is given in such form that blade loading, blade shape, distribution of angular momentum, or distribution of total head may be prescribed. Calculations for single blade rows of aspect ratio 2 and 2/3 are given for a radius ratio of 0.6. They indicate that the process of formation of the axial velocity profile may extend both upstream and downstream of a high-aspect-ratio blade row, while for low aspect ratios the major portion of the three-dimensional flow occurs within the blade row itself. When the through-flow velocity varies greatly from its mean value, the simple linearized solution does not describe the flow process adequately and a more accurate solution applicable to such conditions is suggested. \n\nThe structure of the first-order linearized solution for the axial turbomachine suggested a further approximation employing a minimizing operation. The simplicity of this solution permits the discussion of three interesting problems: Mutual interference of neighboring blade rows in a multistage axial turbomachine, solution for a single blade row of given blade shape, and the solution for this blade row operating at a condition different from the design condition. \n\nIt is found that the interference of adjacent blade rows in the multistage turbomachine may be neglected when the ratio of blade length to the distance between centers of successive blade rows is 1.0 or less. For values of this ratio in excess of 3.0, the interference may be an important influence. The solution for the single blade row indicated that, for the blade shape considered, the distortion of the axial velocity profile caused by off-design operation is appreciably less for low- than for high-aspect-ratio blades. \n\nTo obtain some results for a mixed-flow turbomachine comparable with those for the axial turbomachine as well as to indicate the essential versatility of the method of linearizing the general equations, completely analogous theoretical treatment is given for a turbomachine whose inner and outer walls are concentric cones with common apex and whose flow is that of a three-dimensional source or sink. A particular example for a single rotating blade row is discussed where the angular momentum is prescribed similarly to that used in the examples for the axial turbomachine.", "date": "1952-03", "date_type": "published", "publisher": "National Advisory Committee for Aeronautics", "id_number": "CaltechAUTHORS:MARnacatn2614", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:MARnacatn2614", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "other_numbering_system": { "items": [ { "id": "10", "name": "Guggenheim Jet Propulsion Center Publication" } ] }, "local_group": { "items": [ { "id": "Guggenheim-Jet-Propulsion-Center" } ] }, "primary_object": { "basename": "MARnacatn2614.pdf", "url": "https://authors.library.caltech.edu/records/zt6sp-jyw37/files/MARnacatn2614.pdf" }, "resource_type": "monograph", "pub_year": "1952", "author_list": "Marble, Frank E. and Michelson, Irving" } ]