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"Persamaan Euler merupakan salah satu penyederhanaan persamaan Navier-Stokes dengan asumsi inviscid, adiabatik serta menghilangkan efek dari body force. Pada aliran kompresibel, persamaan Euler merupakan sistem pesamaan hiperbolik non-linear untuk hukum konservasi. Pada aliran kompresibel, munculnya fenomena diskontinuitas berupa gelombang kejut sering menimbulkan masalah dalam simulasi, terutama dalam hal akurasi. Pada skema Godunov, akurasi interpolasi untuk memperoleh fluks pada batas antar sel dapat ditingkatkan dengan penggunaan limiter. Salah satu limiter orde tinggi yang dapat digunakan dalam penyelesaian persamaan Euler adalah skema weighted essentially non-oscillatory (WENO). Masalah yang timbul dari penggunaan skema WENO sebagai limiter adalah beban komputasi yang sangat tinggi, terlebih jika sistem persamaan dan domain komputasi yang kompleks. Pengurangan beban komputasi dapat dilakukan dengan cara simplifikasi skema WENO itu sendiri atau dengan menggunakan skema hibrid dimana skema WENO akan digunakan pada kondisi tertentu. Pada penelitian ini dikembangkan skema hibrid orde tinggi yang mengadopsi WENO pada daerah diskontinu dengan deteksi diskontinuitas secara lokal. Metode cell-centered finite volume digunakan untuk diskretisasi ruang. Penyelesaian masalah Riemann pada batas sel digunakan skema Harten-Lax-van Leer contact (HLLC) dan Lax-Friedrichs, serta untuk integrasi waktu digunakan skema strong stability preserving Runge-Kutta orde ketiga untuk memberikan kestabilan yang baik pada skema numerik. Berdasarkan hasil yang diperoleh, skema hibrid yang dikembangkan cukup efektif digunakan dalam penyelesaian masalah aliran kompresibel. Pengurangan waktu komputasi yang signifikan dan akurasi yang baik menjadikan skema hibrid yang dikembangkan menjadi salah satu pilihan skema numerik orde tinggi yang baik untuk dapat diterapkan dalam simulasi aliran kompresibel.

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Euler equation is a simplification of Navier-Stokes equation which assume the flows are inviscid, adiabatic, and eliminating the effects of body forces. In the compressible flow, the Euler equation is a non-linear hyperbolic conservation laws. The presence of the discontinuities phenomenon in the form of shock wave in the compressible flow often arise the problem in the simulation, mainly in the terms of accuracy. In the Godunovs scheme, the accuracy of interpolation to obtain flux at the intercell boundary can be improved by using a high order limiter. One of the high order limiter that can be used to solve the Euler equation is weighted essentially non-oscillatory (WENO) scheme.

The problem that arises from the use of WENO scheme is high computational loads, moreover the system of equations or the domain are very complex. To reduce the computational cost, it can be done by simplify the WENO reconstruction or implement the hybrid scheme where the WENO scheme only applied in certain conditions.

In this study, hybrid high order scheme are developed which adopt the WENO schem in the discontinuous region by detecting the local discontinuities. The cell-centered finite volume are used in the spatial discretization. Harten-Lax-van Leer contact (HLLC) and Lax-Friedrichs scheme are used to solve Riemann problem in the cell boundary, and third order strong stability preserving Runge-Kutta (SSP-RK) scheme is used for time integration to ensure the positivity and provide good stability in the numerical scheme.

The results shows that the hybrid scheme developed in this work are effective for solving compressible flow problem. The significant reduction of the computational cost and the satisfactory accuracy results are make this hibrid scheme become one of the good choices of high order numerical scheme to be applied in the compressible flow simulation."

"Turunan formula Navier-Stokes dipakai untuk menghitung kerugian tekanan aliran dalam pipa. Panjang pipa, diameter pipa, kecepatan fluida, kekasaran permukaan dan koefisien gesek yang mempengaruhi nilai kerugian tekanan. Formula tersebut tidak berlaku pada belokan/cabang pipa, setelah katup, adanya perubahan diameter (unsteady flow), adanya getaran, dll. Drag reduction dari solusi surfactant atau biopolymer telah menarik perhatian dari sisi konversi energy, dikarenakan penurunan secara mekanis tidak terjadi tetapi menghasilkan drag reduction yang besar di kondisi konsentrasi tertentu oleh karena itu solusi biopolymer banyak digunakan pada system pemipaan dan hasil percobaan dilapangan menunjukan penurunan dari tenaga yang dibutuhkan pompa mencapai 30% dari kecepatan aliran normal Penelitian ini bertujuan untuk membuktikan efek panjang aliran terhadap aliran berkembang penuh dengan membandingkan 3 macam fluida yaitu air murni ; biopolimer air tape ketan 100 ppm dan biopolimer air tape ketan 250 ppm. Di mana biopolimer merupakan hasil fermentasi beras. dan membuktikan keterbatasan penggunaan formula Navier-Stokes. Eksperimen ini menggunakan pipa acrylic berdiameter 12 mm. Variasi panjang pipa masuk terhadap titik pengukur tekanan (pressure tap) yaitu dengan menggeser pipa kecil masuk kedalam pipa uji hingga keadaan fluida mencapai kondisi berkembang penuh. Pada pipa uji dipasang 4 buah pressure tap dengan jarak masing-masing tap 250 mm. Air murni sebagai fluida uji. Debit yang keluar diukur dengan gelas ukur pada periode waktu untuk mendapatkan nilai bilangan Reynolds. Hasil menunjukkan bahwa karakteristik panjang aliran berkembang penuh untuk fluida dengan campuran konsentrasi biopolymer lebih besar dibandingkan dengan air murni.

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Navier-Stokes equations derived formula used to calculate the pressure loss in pipe flow. The length of the pipe, the pipe diameter, the fluid velocity, surface roughness and coefficient of friction that affect the value of pressure loss. The formula does not apply to the branch / branch pipes, the valves, the change in diameter (unsteady flow), the presence of vibration, etc.. Drag reduction of surfactant solution or a biopolymer has attracted attention from the conversion of energy, due to a mechanical reduction does not occur but produced a large drag reduction in a specific concentration of condition therefore biopolymer solutions are widely used in piping systems and field experimental results show a decrease of available power pumps needed to reach 30% of normal flow velocity.

This study aims to prove the effect of flow length to the flow is fully developed by comparing three kinds of fluid that is pure water, 100 ppm biopolymer and 250 ppm biopolymer. Where biopolymer is fermented rice and prove the limitations of the use of the Navier-Stokes formula. These experiments using 12 mm diameter acrylic pipe. Length variation on the point of intake pressure gauge (pressure tap) that is by sliding a small pipe into the test tube until the fluid reaches the state is fully developed conditions. In test tube fitted with a pressure tap four fruit each tap distance 250 mm. Pure water as test fluid. The exit discharge is measured with a measuring cup in the period of time to get the value of Reynolds number. Results showed that the characteristic length for the fluid flow is fully developed with a mixture of biopolymer concentration greater than pure water. "

"The book presents the modern state of the art in the mathematical theory of compressible Navier-Stokes equations, with particular emphasis on the applications to aerodynamics. The topics covered include, modeling of compressible viscous flows, modern mathematical theory of nonhomogeneous boundary value problems for viscous gas dynamics equations, applications to optimal shape design in aerodynamics, kinetic theory for equations with oscillating data, new approach to the boundary value problems for transport equations. "

"Researchers and graduate students in applied mathematics and engineering will find Initial-Boundary Value Problems and the Navier-Stokes Equations invaluable. The subjects addressed in the book, such as the well-posedness of initial-boundary value problems, are of frequent interest when PDEs are used in modeling or when they are solved numerically. The book explains the principles of these subjects. The reader will learn what well-posedness or ill-posedness means and how it can be demonstrated for concrete problems. There are many new results, in particular on the Navier-Stokes equations. When the book was written, the main intent was to write a text on initial-boundary value problems that was accessible to a rather wide audience. Therefore, functional analytical prerequisites were kept to a minimum or were developed in the book. Boundary conditions are analyzed without first proving trace theorems, and similar simplications have been used throughout. The direct approach to the subject still gives a valuable introduction to an important area of applied analysis."

"This second edition, like the first, attempts to arrive as simply as possible at some central problems in the Navier-Stokes equations in the following areas: existence, uniqueness, and regularity of solutions in space dimensions two and three; large time behavior of solutions and attractors; and numerical analysis of the Navier-Stokes equations. Since publication of the first edition of these lectures in 1983, there has been extensive research in the area of inertial manifolds for Navier-Stokes equations. These developments are addressed in a new section devoted entirely to inertial manifolds. Inertial manifolds were first introduced under this name in 1985 and, since then, have been systematically studied for partial differential equations of the Navier-Stokes type. Inertial manifolds are a global version of central manifolds. When they exist they encompass the complete dynamics of a system, reducing the dynamics of an infinite system to that of a smooth, finite-dimensional one called the inertial system. Although the theory of inertial manifolds for Navier-Stokes equations is not complete at this time, there is already a very interesting and significant set of results which deserves to be known, in the hope that it will stimulate further research in this area. These results are reported in this edition. Part I presents the Navier-Stokes equations of viscous incompressible fluids and the main boundary-value problems usually associated with these equations. The case of the flow in a bounded domain with periodic or zero boundary conditions is studied and the functional setting of the equation as well as various results on existence, uniqueness, and regularity of time-dependent solutions are given. Part II studies the behavior of solutions of the Navier-Stokes equation when t approaches infinity and attempts to explain turbulence. Part III treats questions related to numerical approximation. In the Appendix, which is new to the second edition, concepts of inertial manifolds are described, definitions and some typical results are recalled, and the existence of inertial systems for two-dimensional Navier-Stokes equations is shown."