{ "cells": [ { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "# 河床変動の基礎" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "## 動的平衡河道\n", " - まず河床の質量保存則を示す．\n", "$$ \n", "\\begin{align}\n", " (1-\\lambda)\\frac{\\partial z_{b}}{\\partial t}+\\frac{\\partial q_b}{\\partial x}= 0\n", "\\end{align}\n", "$$\n", "\n", "ここに，$\\lambda$は空隙率である．" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "
\n", "\n", "
" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ " - 先程の動画をもう一度見てみよう．" ] }, { "cell_type": "code", "execution_count": 1, "metadata": { "hide_input": true, "slideshow": { "slide_type": "fragment" } }, "outputs": [ { "data": { "image/jpeg": 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"text/html": [ "\n", " \n", " " ], "text/plain": [ "" ] }, "execution_count": 1, "metadata": {}, "output_type": "execute_result" } ], "source": [ "from IPython.display import YouTubeVideo\n", "YouTubeVideo('HOLFmDl4HM4', width=400)" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ " - 土砂は動くが河床高が変わらない状態=動的平衡．河道設計でもっとも重要な概念で目指すべき姿である．安定河道とか平衡河道とかいうものはこの状態を指している．\n", " - 完璧な動的平衡河道ができれば，どんな洪水が起こっても河床変動は0．(=掃流力の大きさと河床変動量は無関係)\n" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "## 河床の移動層厚\n", "\n", " - 動的平衡河道の場合，河床変動は生じないが，土砂が動く層の厚さが掃流力によって変化する．\n", " - 河床変動が生じないため，一般的にあまり議論しないが，現象理解のためには知っておくと便利．\n", " - 式の導出は省略するが，$z=0$における粒子衝突の影響を考慮したせん断力の釣り合いより次式となる．\n" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "\n", "$$\n", "\\dfrac{h_s}{d} = \\dfrac{1}{c_s \\mu_s}(\\tau_* - \\tau_{*c})\n", "$$\n", "\n", "ここに，$c_s$は静止堆積濃度\n", "
\n", " \n", "
" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "## 掃流砂量式(掃流砂量関数)\n", "\n", " - 連続体で評価した掃流砂量を示す関数である．\n", " - ほぼ経験的な式で多くの式が提案されている．例外はあるが概ね以下のような形になっている．\n", " - *どの式も不完全であり万能ではない*\n", " " ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "$$\n", " \\dfrac{q_b}{\\sqrt{sgd^3}} = \\alpha \\tau_*^\\beta \\left(1-\\dfrac{\\tau_{*c}}{\\tau_*} \\right)^\\gamma \\left(1-\\sqrt{\\dfrac{\\tau_{*c}}{\\tau_*} }\\right)^\\delta\n", "$$\n", " \n", " - Ashida-Michiue:$\\alpha= 17,\\beta=3/2,\\gamma=1,\\delta=1$\n", " - MPM:$\\alpha= 8,\\beta=3/2,\\gamma=3/2,\\delta=0$\n", " - Brown:$\\alpha= 10,\\beta=5/2,\\gamma=0,\\delta=0$\n", " - Sato-Kikkawa-Ashida:$\\alpha= {\\rm func},\\beta=3/2,\\gamma=0,\\delta=0$" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "# 浮遊砂量式（詳細は省略）\n", " - 土砂濃度の拡散方程式について，定常・等流場を想定し（x,y微分項はキャンセル），河床近傍から水面までの積分すると次式が得られる．\n", "\n", "$$ \n", "\\begin{align}\n", " \\dfrac{\\bar{C}}{C_a} &= \\left( \\dfrac{h-z}{z}\\dfrac{z_a}{h-z_a}\\right)^Z \\\\\n", " Z &= \\dfrac{w_0}{\\kappa u_*}\n", "% (1-\\lambda)\\frac{\\partial z_{b}}{\\partial t}+\\frac{\\partial q_b}{\\partial x} &= 0 \\\\\n", "% q_b &= {\\rm func} (\\tau_*)\n", "\\end{align}\n", "$$" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "ここに，$\\bar{C}$:水深平均濃度，$z_a$は河床近傍の任意箇所（水深の5%と定義)，$C_a$:$z_a$の濃度で基準面濃度と呼ばれる．\n", "\n", "\n", "\n", "
\n", " \n", "
" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ " - 水深方向に分布を持ち，その形状は$u_*/w_0$によって決まる．この関係を実験値と比較したもの下図である．\n", " \n", "
\n", "\n", "
\n", "「土砂水理学1」より引用\n", " " ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ " - 基準面濃度$C_a$が決まれば濃度分布が決まる．基準面濃度は河床からの巻き上げ量によって決まる．\n", " - 浮遊砂量式とは，基準面濃度を示すものであり，様々な形の経験則が提案されている．詳細は専門書を参考されたい．" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "\n", " - 浮遊砂は，かつては沖積河川の河床変動に寄与しないため，ダム堆砂，感潮区間を除いてはそれほど重要視されなかった．しかし，現在では高水敷形成に大きく寄与していると考えられており，高水敷の河床変動を考える上では重要．\n", " - 流れの三次元性に強く影響を受けるため，流れ場の解析技術がもう少し発展しないと高精度の解析は難しいと考えられる．\n" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "# 河床変動計算\n", "\n", "これまでの内容を基に河床変動計算をやってみよう．支配方程式はたったこれだけ.\n", "$$ \n", "\\begin{align}\n", " (1-\\lambda)\\frac{\\partial z_{b}}{\\partial t}+\\frac{\\partial q_b}{\\partial x} &= 0 \\\\\n", " q_b &= {\\rm func} (\\tau_*)\n", "\\end{align}\n", "$$" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "## case1:河床掘削後の応答\n", "\n", " - 縦断距離1km，河床勾配1/1000，粒径4mm．100mの区間を10cm掘削した場合の応答\n" ] }, { "cell_type": "code", "execution_count": 2, "metadata": { "hide_input": true, "slideshow": { "slide_type": "slide" } }, "outputs": [ { "data": { "text/html": [ "\n", " \n", " " ], "text/plain": [ "" ] }, "execution_count": 2, "metadata": {}, "output_type": "execute_result" } ], "source": [ "from IPython.display import IFrame\n", "IFrame(\"fig/case1.html\",width=800,height=350)" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ " - 局所的な掘削箇所は必ず元に戻る．" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "## case2:縦断変化\n", "\n", " - 縦断距離10km，河床勾配～5km:1/200，5km～：1/1000，粒径4mm．変曲点のある河道の変化" ] }, { "cell_type": "code", "execution_count": 3, "metadata": { "hide_input": false, "slideshow": { "slide_type": "slide" } }, "outputs": [ { "data": { "text/html": [ "\n", " \n", " " ], "text/plain": [ "" ] }, "execution_count": 3, "metadata": {}, "output_type": "execute_result" } ], "source": [ "from IPython.display import IFrame\n", "IFrame(\"fig/case2.html\",width=800,height=350)" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ " - 河床変動が進むと上流から下流を結んだ必ず直線になる．\n", "\n" ] } ], "metadata": { "celltoolbar": "Slideshow", "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.6.10" }, "toc": { "base_numbering": 1, "nav_menu": {}, "number_sections": true, "sideBar": true, "skip_h1_title": false, "title_cell": "Table of Contents", "title_sidebar": "Contents", "toc_cell": false, "toc_position": {}, "toc_section_display": true, "toc_window_display": false } }, "nbformat": 4, "nbformat_minor": 2 }