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On the other hand, this system is believed to be governed by the NavierStokes equations which consist of the equations of motion and of continuity. However, relationships between these equations and phenomenologies on statistical properties of smallscale motions have not been clearly understood primarily because such a statistical theory is hard to construct due to the nonlinearity of the basic equations. Since the nonlinearity causes an infinite hierarchy of moments, we never obtain a closed set of equations for a finite number of statistical quantities without any assumptions. This is the socalled closure problem in the statistical theory of turbulence. We adopt the directinteraction approximation (DIA), which was originally proposed by Kraichnan (1959), to attack and solve the closure problem.\u003cbr /\u003e The DIA is an excellent approximation in the sense that the nonlinearity is never neglected and no adjustable parameter is introduced in the formulation. Unfortunately, however, it is known that a closed set of equations obtained by a naive application of DIA (Kraichnan 1959) to the NavierStokes system is inconsistent with experimental observation. Especially, it is E(k) α k3/2 that the closure equations predict as the energy spectrum E(k) in the inertial range, where the k5/3 power law is observed by many experiments. This inconsistency implies incompleteness of the application of DIA to the NavierStokes system. Although Kraichnan (1965) improved the application method of DIA and succeeded in deriving the k5/3 power law, the formulations are too complicated to be justified. Moreover, in spite of its long history and important role in the field of the statistical theory of turbulence, the essence of DIA may have been misunderstood by many researchers. This is due to the fact that validity conditions and applicability of DIA were not clear.\u003cbr /\u003e We introduce a model equation, consisting of quadratic nonlinear and linear dissipative terms, which is simpler than the NavierStokes equation but still possesses its important mathematical structures. Then, it is shown that DIA is valid for such a system that has weak nonlinear couplings and large numbers of degrees of freedom even if nonlinearity of the system is strong (i.e., the nonlinear terms are larger than the linear ones in magnitude). Furthermore, we clarify similarities and differences between DIA and a Reynoldsnumber expansion socalled RRE (Reynoldsnumber reversed expansion) . For some known systems, including the NavierStokes system and the present model, these two approximations yield an identical set of equations for the correlation and the response functions. Owing to this fact, these two approximations have sometimes been identified erroneously. It must be stressed, however, that DIA and RRE are based upon completely different ideas and working assumptions. Hence, we should distinguish these two theories. This is reasonable because the validity conditions of DIA depend on the strength of nonlinear couplings and the number of degrees of freedom, but not on the Reynolds number, while the validity of RRE depends crucially on magnitude of the Reynolds number.\u003cbr /\u003e We further investigate the validity condition of DIA and the relationships between DIA and RRE from a viewpoint of the strength of nonlinear couplings by extending the model equation. It is then shown that DIA is valid for systems such that the average number of direct interactions between a pair of modes is much smaller than the square root of the number of degrees of freedom, and that RRE may be regarded as an approximation under which the nonlinear terms are replaced by a jointGaussian random variables. The last approximation, called normal nonlinear term approximation, has the same validity conditions as DIA.\u003cbr /\u003e Smallscale motions of turbulence may be statistically homogeneous, and the number of degrees of freedom of this system increases in proportion to the 9/4 power of the Reynolds number. Hence, smallscale motions of turbulent fields at high Reynolds number satisfy the two validity conditions of DIA, i.e., weakness of nonlinear couplings and largeness of the degrees of freedom. This implies that DIA is applicable to this system. As mentioned above, however, when we apply DIA to the Eulerian velocity correlation function and the Eulerian velocity response function (Kraichnan 1959), we encounter the difficulty that the resultant closure equations are incompatible with experiments. Here, we instead apply DIA to the Lagrangian velocity correlation function and the Lagrangian response function with the help of the position function (Kaneda 1981), which is a map between the Eulerian and the Lagrangian fields. The resultant equations yield not only the wellknown k5/3 power law predicted phenomenologically by Kolmogorov (1941) of the energy spectrum, but also the functional form in the entire universal range, which excellently agrees with experimental data.\u003cbr /\u003e We next apply DIA to passive scalar fields (temperature, particle concentration, smoke, and so on) advected by turbulence without affecting fluid motions. Then it is systematically shown that solutions to the resultant closure equations by DIA for the Lagrangian correlation and the response functions for the velocity and the passive scalar fields are completely consistent with the phenomenologies on the scalar spectrum by Obukhov (1949) and Corrsin (1951) in the inertialadvective range, Batchelor, Howells \u0026 Townsend (1959) in the inertialdiffusive range, and Batchelor (1959) in the viscousadvective range.", "subitem_description_type": "Other"}]}, "item_1_description_18": {"attribute_name": "フォーマット", "attribute_value_mlt": [{"subitem_description": "application/pdf", "subitem_description_type": "Other"}]}, "item_1_description_7": {"attribute_name": "学位記番号", "attribute_value_mlt": [{"subitem_description": "総研大甲第394号", "subitem_description_type": "Other"}]}, "item_1_select_14": {"attribute_name": "所蔵", "attribute_value_mlt": [{"subitem_select_item": "有"}]}, "item_1_select_8": {"attribute_name": "研究科", "attribute_value_mlt": [{"subitem_select_item": "数物科学研究科"}]}, "item_1_select_9": {"attribute_name": "専攻", "attribute_value_mlt": [{"subitem_select_item": "10 核融合科学専攻"}]}, "item_1_text_10": {"attribute_name": "学位授与年度", "attribute_value_mlt": [{"subitem_text_value": "1998"}]}, "item_1_text_20": {"attribute_name": "業務メモ", "attribute_value_mlt": [{"subitem_text_value": "（2018年2月9日）本籍など個人情報の記載がある旧要旨・審査要旨を個人情報のない新しいものに差し替えた。承諾書等未確認。要確認該当項目修正のこと。"}]}, "item_creator": {"attribute_name": "著者", "attribute_type": "creator", "attribute_value_mlt": [{"creatorNames": [{"creatorName": "GOTO, Susumu", "creatorNameLang": "en"}], "nameIdentifiers": [{"nameIdentifier": "8605", "nameIdentifierScheme": "WEKO"}]}]}, "item_files": {"attribute_name": "ファイル情報", "attribute_type": "file", "attribute_value_mlt": [{"accessrole": "open_date", "date": [{"dateType": "Available", "dateValue": "20160217"}], "displaytype": "simple", "download_preview_message": "", "file_order": 0, "filename": "甲394_要旨.pdf", "filesize": [{"value": "406.5 kB"}], "format": "application/pdf", "future_date_message": "", "is_thumbnail": false, "licensetype": "license_11", "mimetype": "application/pdf", "size": 406500.0, "url": {"label": "要旨・審査要旨 / Abstract, Screening Result", "url": "https://ir.soken.ac.jp/record/476/files/甲394_要旨.pdf"}, "version_id": "da2bd21d8d6a4b74adb639f912f7c35d"}, {"accessrole": "open_date", "date": [{"dateType": "Available", "dateValue": "20160217"}], "displaytype": "simple", "download_preview_message": "", "file_order": 1, "filename": "甲394_本文.pdf", "filesize": [{"value": "16.1 MB"}], "format": "application/pdf", "future_date_message": "", "is_thumbnail": false, "licensetype": "license_11", "mimetype": "application/pdf", "size": 16100000.000000002, "url": {"label": "本文", "url": "https://ir.soken.ac.jp/record/476/files/甲394_本文.pdf"}, "version_id": "02e539e275b0489d86d5f1b6fca2a2e5"}]}, "item_language": {"attribute_name": "言語", "attribute_value_mlt": [{"subitem_language": "eng"}]}, "item_resource_type": {"attribute_name": "資源タイプ", "attribute_value_mlt": [{"resourcetype": "thesis", "resourceuri": "http://purl.org/coar/resource_type/c_46ec"}]}, "item_title": "DirectInteraction Approximation  Principles andApplications", "item_titles": {"attribute_name": "タイトル", "attribute_value_mlt": [{"subitem_title": "DirectInteraction Approximation  Principles andApplications"}, {"subitem_title": "DirectInteraction Approximation  Principles andApplications", "subitem_title_language": "en"}]}, "item_type_id": "1", "owner": "1", "path": ["12"], "permalink_uri": "https://ir.soken.ac.jp/records/476", "pubdate": {"attribute_name": "公開日", "attribute_value": "20100222"}, "publish_date": "20100222", "publish_status": "0", "recid": "476", "relation": {}, "relation_version_is_last": true, "title": ["DirectInteraction Approximation  Principles andApplications"], "weko_shared_id": 1}
DirectInteraction Approximation  Principles andApplications
https://ir.soken.ac.jp/records/476
https://ir.soken.ac.jp/records/476aeb3ba6158894c60984f9e10c46f9156
名前 / ファイル  ライセンス  アクション 

要旨・審査要旨 / Abstract, Screening Result (406.5 kB)


本文 (16.1 MB)

Item type  学位論文 / Thesis or Dissertation(1)  

公開日  20100222  
タイトル  
タイトル  DirectInteraction Approximation  Principles andApplications  
タイトル  
言語  en  
タイトル  DirectInteraction Approximation  Principles andApplications  
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_46ec  
資源タイプ  thesis  
著者名 
後藤, 晋
× 後藤, 晋 

フリガナ 
ゴトウ, ススム
× ゴトウ, ススム 

著者 
GOTO, Susumu
× GOTO, Susumu 

学位授与機関  
学位授与機関名  総合研究大学院大学  
学位名  
学位名  博士（理学）  
学位記番号  
内容記述タイプ  Other  
内容記述  総研大甲第394号  
研究科  
値  数物科学研究科  
専攻  
値  10 核融合科学専攻  
学位授与年月日  
学位授与年月日  19990324  
学位授与年度  
1998  
要旨  
内容記述タイプ  Other  
内容記述  It has been phenomenologically shown and widely supported by experiments that statistical properties in small scales of turbulence of incompressible fluids exhibit some universality irrespective of the kinds of fluids, boundary condition and the Reynolds number. On the other hand, this system is believed to be governed by the NavierStokes equations which consist of the equations of motion and of continuity. However, relationships between these equations and phenomenologies on statistical properties of smallscale motions have not been clearly understood primarily because such a statistical theory is hard to construct due to the nonlinearity of the basic equations. Since the nonlinearity causes an infinite hierarchy of moments, we never obtain a closed set of equations for a finite number of statistical quantities without any assumptions. This is the socalled closure problem in the statistical theory of turbulence. We adopt the directinteraction approximation (DIA), which was originally proposed by Kraichnan (1959), to attack and solve the closure problem.<br /> The DIA is an excellent approximation in the sense that the nonlinearity is never neglected and no adjustable parameter is introduced in the formulation. Unfortunately, however, it is known that a closed set of equations obtained by a naive application of DIA (Kraichnan 1959) to the NavierStokes system is inconsistent with experimental observation. Especially, it is E(k) α k3/2 that the closure equations predict as the energy spectrum E(k) in the inertial range, where the k5/3 power law is observed by many experiments. This inconsistency implies incompleteness of the application of DIA to the NavierStokes system. Although Kraichnan (1965) improved the application method of DIA and succeeded in deriving the k5/3 power law, the formulations are too complicated to be justified. Moreover, in spite of its long history and important role in the field of the statistical theory of turbulence, the essence of DIA may have been misunderstood by many researchers. This is due to the fact that validity conditions and applicability of DIA were not clear.<br /> We introduce a model equation, consisting of quadratic nonlinear and linear dissipative terms, which is simpler than the NavierStokes equation but still possesses its important mathematical structures. Then, it is shown that DIA is valid for such a system that has weak nonlinear couplings and large numbers of degrees of freedom even if nonlinearity of the system is strong (i.e., the nonlinear terms are larger than the linear ones in magnitude). Furthermore, we clarify similarities and differences between DIA and a Reynoldsnumber expansion socalled RRE (Reynoldsnumber reversed expansion) . For some known systems, including the NavierStokes system and the present model, these two approximations yield an identical set of equations for the correlation and the response functions. Owing to this fact, these two approximations have sometimes been identified erroneously. It must be stressed, however, that DIA and RRE are based upon completely different ideas and working assumptions. Hence, we should distinguish these two theories. This is reasonable because the validity conditions of DIA depend on the strength of nonlinear couplings and the number of degrees of freedom, but not on the Reynolds number, while the validity of RRE depends crucially on magnitude of the Reynolds number.<br /> We further investigate the validity condition of DIA and the relationships between DIA and RRE from a viewpoint of the strength of nonlinear couplings by extending the model equation. It is then shown that DIA is valid for systems such that the average number of direct interactions between a pair of modes is much smaller than the square root of the number of degrees of freedom, and that RRE may be regarded as an approximation under which the nonlinear terms are replaced by a jointGaussian random variables. The last approximation, called normal nonlinear term approximation, has the same validity conditions as DIA.<br /> Smallscale motions of turbulence may be statistically homogeneous, and the number of degrees of freedom of this system increases in proportion to the 9/4 power of the Reynolds number. Hence, smallscale motions of turbulent fields at high Reynolds number satisfy the two validity conditions of DIA, i.e., weakness of nonlinear couplings and largeness of the degrees of freedom. This implies that DIA is applicable to this system. As mentioned above, however, when we apply DIA to the Eulerian velocity correlation function and the Eulerian velocity response function (Kraichnan 1959), we encounter the difficulty that the resultant closure equations are incompatible with experiments. Here, we instead apply DIA to the Lagrangian velocity correlation function and the Lagrangian response function with the help of the position function (Kaneda 1981), which is a map between the Eulerian and the Lagrangian fields. The resultant equations yield not only the wellknown k5/3 power law predicted phenomenologically by Kolmogorov (1941) of the energy spectrum, but also the functional form in the entire universal range, which excellently agrees with experimental data.<br /> We next apply DIA to passive scalar fields (temperature, particle concentration, smoke, and so on) advected by turbulence without affecting fluid motions. Then it is systematically shown that solutions to the resultant closure equations by DIA for the Lagrangian correlation and the response functions for the velocity and the passive scalar fields are completely consistent with the phenomenologies on the scalar spectrum by Obukhov (1949) and Corrsin (1951) in the inertialadvective range, Batchelor, Howells & Townsend (1959) in the inertialdiffusive range, and Batchelor (1959) in the viscousadvective range.  
所蔵  
値  有  
フォーマット  
内容記述タイプ  Other  
内容記述  application/pdf 