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Theory of secondary flow in cascades
https://doi.org/10.18999/memfenu.29.2.165
https://doi.org/10.18999/memfenu.29.2.165e29d407c-5cf8-4ee5-a7ad-ac44c851b50d
| 名前 / ファイル | ライセンス | アクション |
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29-2-02.pdf (4.8 MB)
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| Item type | 紀要論文 / Departmental Bulletin Paper(1) | |||||
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| 公開日 | 2020-11-24 | |||||
| タイトル | ||||||
| タイトル | Theory of secondary flow in cascades | |||||
| 言語 | en | |||||
| 著者 |
Otsuka, Shintaro
× Otsuka, Shintaro |
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| アクセス権 | ||||||
| アクセス権 | open access | |||||
| アクセス権URI | http://purl.org/coar/access_right/c_abf2 | |||||
| 抄録 | ||||||
| 内容記述タイプ | Abstract | |||||
| 内容記述 | It was in 1951 that Squire & Winter published a new conception that the secondary flow could occur in a bend through which a perfect fluid is flowing, as a result of a non-uniform distribution of velocity at entrance to the bend. Their idea was reformed by Hawthorne and qualitative successes were gained on the flow in duct bend and the flow around strut placed in an approaching stream with a nonuniform velocity (such as a bridge pier in a river). Although the first attempt of Squire & Winter is supposed to take aim at the solution of cascade secondary flows, the application of this idea to blade rows (turbo-machine cascade and linear cascade) has almost eintirely been unsuccessful. The reason of this can be divided into two parts. The first is the assumption of inviscid perfect fluid assumed by Squire & Winter and viscosity neglected, and the second is that the theory itself had some imperfectness. The opinion, that the reason why the secondary flow theory is not useable to the problem of cascade simply only because of the assumption of perfect fluid without viscosity, is considered by the author to be unsatisfactory. The author wants to consider the imperfectness of theory itself prior to the effect of viscosity. After mending the imperfectness of theory there comes a step of saying the effect of viscosity. Contributors to the secondary flow theory other than Squire & winter and Hawthorne mentioned above must be L. H. Smith Jr. and Wislicenus. Smith rendered a distinguished service in verifying clearly that the vortex components in the exit flow of cascade are consisted of the trailing vortices which are formed by a component concerning the vorticity in the upstream and a component corresponding to the variation of blade circulation, in addition to the passage vortex (vorticity) which is formed by the deformation of the upstream vorticity through the blade passage. If we rearrange his results, we have 〔vortex component normal to the cascade exit flow〕 (1) normal component to the flow of the passage vortex (vorticity). 〔vortex component parallel to the cascade exit flow〕 (1)’ parallel component to the flow of the passage vortex. (2)’ trailing filament vortex (component of trailing vortex concerning the vorticity in the upstream). (3)’ trailing shed vortex (component of trailing vortex corresponding the variation of blade circulation). In spite of his important contribution mentioned above he made a regrettable mistake in the henceforth calculation of secondary vortex. The mistake made by him is a point of problem itself of secondary flow theory and will be considered later. And we need some elucidation about the method of solution prior to the explanation of the mistake. At first we must define what is the secondary flow. Generally the secondary flow is defined as the difference of the ideal flow to the actual flow which differs from the former by the existence of boundary layer in the flow. But this definition is inconvenient and ambiguous since the solution differs in accordance with the definition of “ideal” flow. But the situation is left as it is, and this might be the root of the mistake such as Smith’s. The most intelligible definition of secondary flow may be the one starting from the secondary vortex. If the vortex is contained in the flow, the streamwise component of this vortex is called the secondary vortex, and the flow induced by the secondary vortex is the secondary flow. The ideal flow mentioned above, therefore, is the one containing no streamwise vortex, and the irrotational flow will be conveniently accepted. This idea is especially expedient for obtaining the first approximation of secondary flow. Namely, the method of solution is to find what attitude will be taken by the vortex which was originally contained in the upstream and drifted with the fluid into the downstream. This is the means often used in the treatment of secondary flow in linear cascade. (see Fig. 7-1) But his idea is not convenient for the treatment of secondary flows in turbomachine. When the flow in a turbomachine is irrotational, it is of so-called free-vortex type. But the flow in turbomachine is often different from the free-vortex type, and therefore it may not be advisable to take the free-vortex type of flow as the zeroth approximation. Generally we choose the axisymmetric flow as the zeroth approximation in this occasion. (The flow around two-dimensional cascade is employed so as to approximate the flow near blade row.) But we must notice here that in ordinary cases the axisymmetric flow has vorticity in it. Therefore, we should think that the secondary flow is contained in this flow, which means we should have sufficient knowledges of the behavior of vortex in this flow. There are two understandings on this axisymmetric flow corresponding to the zeroth approximation, that it is a flow of ideal fluid and has no boundary layer in the inlet side, and then the effect of boundary layer and finite blade spacing are caught as the secondary flow,......, or it is an axisymmetric vortex flow containing the boundary layer in it and the variation of flow caused by the finiteness of blade pitch is caught as the secondary flow ...... Both will be of use, and it is easy to understand that the knowledge of vortices in this axisymmetric flow should be indispensable for the solution. But the situation was not so. Smith’s mistake mentioned above is that the axisymmetric flow considered by him was not the true one, and if sufficient considerations had been done his miscarriage would be avoided. The examination of axisymmetric flow had some curious difficulties in spite of its simple appearances. Namely, the ordinary method of solution of axisymmetric flow has no information of the stream-wise component of vortex (secondary vortex) in the exit flow of cascade. This was pointed out by Wislicenus for the first time. In fine the answer cannot be obtained from the ordinary axisymmetric theory when we want the solution from the standpoint of secondary flows. Even in the axisymmetric theory the streamwise component of passage vortex and the trailing filament vortex can be easily obtained by assuming the flow in blade passage (of infinitesimal spacing). (The sum of the both can be obtained without assuming its flow.) The problem is the trailing shed vortex, and a few trials were done without any success. Because the flow pattern of the exit flow of blade row can be obtained from the axisymmetric theory, the author tried to calculate the vorticity in it (which means he calculates the secondary vorticity) and found that it could be easily obtainable. The fact which is most important and curious in the results is that when the exit angle of blade row is of free vortex type i. e. tan γ_2e=C⁄r (where γ_2e : flow angle at the exit of blade row, C : a constant, r : radial position), there is no streamwise vorticity or secondary vorticity in the downstream whatever vortices are contained in the upstream. Although this fact was pointed out by Preston, this phenomenon which can be named as vortex rectification may be interesting and important. This fact has a great meaning when we employ an axisymetric flow as a flow of the zeroth approximation (base flow). In other words, because there is no secondary flow in the downstream of base flow of free vortex type no matter how the condition of the upstream may be, we can get the three-dimensional flow of the blade row of finite blade spacing if we add the secondary flow correspoing to the finite blade spacing to this base flow. The data of the secondary flow of linear cascades will be useful for this process. If the flow is other than the free vortex type the method of treatment is not clear, but the ideas mentioned above may become good references. In the next place, the calculation of secondary velocities from secondary vortices will be done by Hawthorne’s propositions. Hawthorne set the following assumptions, [1] the secondary flow occurs in planes which are normal to the average stream direction, (Trefftz plane), [2] the secondary vorticity is normal to these planes, and [3] the secondary flows may be treated as a two-dimensional plane flow superimposed on the main flow. If we represent these about the linear cascade, we have Fig. 7-2. The treatment in regard to Trefftz plane has a great defect that although the solution as two-dimensional flows is possible if this Trefftz plane can be considered being a plane, but nothing can be done when the flow in a turbomachine should be treated. The Hawthorne’s propositions, therefore, can be effectively applicable only to the linear cascade. When we want to get secondary velocities in linear cascade such as illustrated in Fig. 7-2, the shape of boundary of Trefftz plane should be a problem. In the figure the boundary is illustrated as a rectangle, but even though AB is straight we must examine whether CD and other vortex sheets are straight or not. These have been proved to be straight. In fine the secondary flow in exit flow of linear cascade can be obtained from the calculation of flow in a rectangle ABDC which has vorticities ω_2ps in it. Induced velocities in x-direction induced at AB or CD form the trailing vortex sheet which is the sum of the trailing filament vortex and the trailing shed vortex. Now, we must be careful that there exists an assumption of great importance in the above ideas. Consideration of the secondary flow in a rectangle means that we accept the idea that there is no flow in AC-direction (y-direction) at the trailing edge AB, and there remains some doubt that the Kutta’s condition of blade in the flow containing secondary flows can be expressed by the above or not. Probably it may be accepted when the trailing edge is very thin or of cusped form. And if we accept the above assumption, we can reach a noteworthy conclusion that the direction of the wake (vortex sheet) will show the exit flow direction of two-dimensional cascade (i. e. the cascade containing no secondary flow), because sides AB and CD are not deformed by the secondary flow. The exit flow angle of two-dimensional cascade should be obtained immediately from the direction of wake without a troublesome method such as the boundary layer suction! This is going to be proved experimentally. (not yet published) Now let us consider again on the component of vortices in the downstream of linear cascade. (1) corresponds to the boundary layer in the downstream and we can explain the phenomenon such as the development of boundary layers in the decelerating cascade. (2)’+(3)’ forms the trailing vortex and can be expressed by spanwise velocities along upper and lower sides of the rectangle. (1)’+(2)’ can be obtained from the idea that the passage vortex is connected through the wake (its streamwise component). This is not related to the cascade configurations but related only to inflow and outflow directions. In fine (1)’+(2)’ is the secondary vortex appearing from the phenomenon that the flow is turned. (3)’ is the trailing vortex component corresponding to the variation of blade circulation which has close relation to the cascade configuration, as against (1)’+(2)’ being not related to those. We can recognize what characterizes the secondary flow in cascade (that is, what characterizes the turning of flow under the existence of blades) is the variation of blade circulation and the trailing vortex accompanied by it. As aforesaid the solution of secondary flow in linear cascade seems to be obtained from the calculation of the flow in the rectangle, but even though the boundary is maintained rectangular the averaged flow angle of flow is changed from the direction of wake (may be the directoin of two-dimensional case) by the flow inside. We can find the turning angle is smaller (under turning) at the center of span, larger (over turning) at side wall, and smallest at the border between boundary layer and main flow. These explain qualitatively the result of cascade experiment to some extent. Further studies must be needed on the effect of fluid viscosity. | |||||
| 言語 | en | |||||
| 出版者 | ||||||
| 出版者 | Faculty of Engineering, Nagoya University | |||||
| 言語 | en | |||||
| 言語 | ||||||
| 言語 | eng | |||||
| 資源タイプ | ||||||
| 資源タイプ識別子 | http://purl.org/coar/resource_type/c_6501 | |||||
| 資源タイプ | departmental bulletin paper | |||||
| 出版タイプ | ||||||
| 出版タイプ | VoR | |||||
| 出版タイプResource | http://purl.org/coar/version/c_970fb48d4fbd8a85 | |||||
| ID登録 | ||||||
| ID登録 | 10.18999/memfenu.29.2.165 | |||||
| ID登録タイプ | JaLC | |||||
| ISSN（print） | ||||||
| 収録物識別子タイプ | PISSN | |||||
| 収録物識別子 | 0027-7657 | |||||
| 書誌情報 |
en : Memoirs of the Faculty of Engineering, Nagoya University 巻 29, 号 2, p. 165-231, 発行日 1978-03-31 |
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| 著者版フラグ | ||||||
| 値 | publisher | |||||
