{ "cells": [ { "cell_type": "code", "execution_count": 10, "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [ "remove-input" ] }, "outputs": [], "source": [ "import numpy as np\n", "import matplotlib.pyplot as plt\n", "%config InlineBackend.rc = {'figure.dpi': 300}\n", "%config InlineBackend.figure_format = 'svg'\n", "import sys\n", "sys.path.append(\"../py_figures\")\n", "import diagrams\n", "from myst_nb import glue\n", "import inter_fig_chap3" ] }, { "cell_type": "markdown", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "source": [ "(content:chap3:flow-over-step)=\n", "# Flow over a step in the aquifer base\n", "The problem of unconfined flow between two parallel rivers without areal recharge is considered, but this time, there is a step in the base of the aquifer as shown in {numref}`Figure %s `. The base of the aquifer is equal to $z_0$ for the left part of the aquifer (length $L_0$) and is equal to $z_1$ for the right part of the aquifer (length $L_1$). The head is $h_l$ at $x =0$, and the head is $h_r$ at $x=L$. These conditions are transformed to conditions in terms of the discharge potential\n", "\n", "$$\n", "\\Phi\\vert_{x=0}=\\Phi_l=\\tfrac{1}{2}k(h_l-z_0)^2\n", "$$(eq_cond1_3.2)\n", "\n", "$$\n", "\\Phi\\vert_{x=L}=\\Phi_r=\\tfrac{1}{2}k(h_r-z_1)^2\n", "$$(eq_cond2_3.2)\n", "\n", "As for the solution of, e.g., {numref}`content:chap1:flow-two-zones`, the head and flux are continuous at $x=L_0$\n", "\n", "$$\n", "h^-=h^+\n", "$$(eq_sol1_3.2)\n", "\n", "$$\n", "Q_x^-=Q_x^+\n", "$$(eq_sol2_3.2)\n", "\n", "where the superscripts $^-$ and $^+$ are used as shorthand for evaluation just to the left and right of $x=L_0$, respectively. " ] }, { "cell_type": "code", "execution_count": 3, "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [ "remove-input", "remove-output" ] }, "outputs": [ { "data": { "application/papermill.record/image/svg+xml": "\n\n\n", "application/papermill.record/text/plain": "

" }, "metadata": { "scrapbook": { "mime_prefix": "application/papermill.record/", "name": "fig_stepbase3_2" } }, "output_type": "display_data" }, { "data": { "image/svg+xml": [ "\n", "\n", "\n" ], "text/plain": [ "

" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "fig = diagrams.solution3_2()\n", "glue(\"fig_stepbase3_2\", fig, display=False)" ] }, { "cell_type": "markdown", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "source": [ "```{glue:figure} fig_stepbase3_2 \n", "---\n", "name: fig_stepbase3.2\n", "---\n", "Unconfined flow with a step in the base.\n", "```" ] }, { "cell_type": "markdown", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "source": [ " The flow from the left river to the right river is equal to $U$, which is unknown upfront. Application of boundary conditions ({eq}`eq_cond1_3.2`) and ({eq}`eq_cond2_3.2`) gives equations for the discharge potential for the left and right parts that still contain the unknown flux $U$\n", "\n", "$$\n", "\\Phi=-Ux + \\Phi_l \\hskip 2em 0 \\le x \\le L_0\n", "$$(eq_lefpart_3.2)\n", "\n", "$$\n", "\\Phi=-U(x-L) + \\Phi_r \\hskip 2em L_0 \\le x \\le L\n", "$$(eq_rigpart_3.2)\n", "\n", "where $L=L_0+L_1$. The discharge $U$ is obtained from condition ({eq}`eq_sol1_3.2`) as follows. The discharge potentials $\\Phi^-$ and $\\Phi^+$ at $x=L_0$ are\n", "\n", "$$\n", "\\Phi^- = -UL_0 + \\Phi_l = \\tfrac{1}{2}k(h^- -z_0)^2\n", "$$(eq_phineg_3.2) \n", "\n", "$$\n", "\\Phi^+ = -U(L_0 - L) + \\Phi_r = \\tfrac{1}{2}k(h^+ -z_1)^2\n", "$$(eq_phipos_3.2)\n", "\n", "which are transformed in the following expressions for $h^-$ and $h^+$\n", "\n", "$$\n", "h^- = \\sqrt{\\frac{2(-UL_0+\\Phi_l)}{k}} + z_0\n", "$$(eq_hneg_3.2)\n", "\n", "$$\n", "h^+ = \\sqrt{\\frac{2(UL_1+\\Phi_r)}{k}} + z_1\n", "$$(eq_hpos_3.2)\n", "\n", "Setting $h^-=h^+$ gives an equation from which $U$ can be determined, but it is not possible to write an explicit equation for $U$. The value of $U$ for which $h^-=h^+$ is determined numerically. First, $h^-$ and $h^+$ are plotted vs. $U$ in {numref}`Figure %s `." ] }, { "cell_type": "code", "execution_count": 12, "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "outputs": [], "source": [ "# parameters\n", "k = 10 # hydraulic conductivity, m/d\n", "z0 = 0 # base elevation left section, m\n", "z1 = -4 # base elevation right section, m\n", "L0 = 500 # length of left section, m\n", "L1 = 500 # length of right section, m\n", "L = L0 + L1 # total distance between rivers, m\n", "hl = 10 # specified head at the left boundary, m\n", "hr = 0 # specified head at the right boundary, m" ] }, { "cell_type": "code", "execution_count": 14, "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "outputs": [], "source": [ "# solution\n", "phil = 0.5 * k * (hl - z0)**2\n", "phir = 0.5 * k * (hr - z1)**2\n", "\n", "def hmin(U, L0=L0, z0=z0, phil=phil):\n", " return np.sqrt(2 * (-U * L0 + phil) / k) + z0\n", "\n", "def hplus(U, L1=L1, z1=z1, phir=phir):\n", " return np.sqrt(2 * (U * L1 + phir) / k) + z1" ] }, { "cell_type": "code", "execution_count": 16, "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [ "remove-output" ] }, "outputs": [ { "data": { "image/svg+xml": [ "\n", "\n", "\n" ], "text/plain": [ "

" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# basic plot two conditions\n", "U = np.linspace(0, 1, 100)\n", "plt.plot(U, hmin(U), label='$h^-$')\n", "plt.plot(U, hplus(U), label='$h^+$')\n", "plt.legend();" ] }, { "cell_type": "code", "execution_count": 9, "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [ "remove-input", "remove-output" ] }, "outputs": [ { "data": { "application/papermill.record/image/svg+xml": "\n\n\n", "application/papermill.record/text/plain": "

" }, "metadata": { "scrapbook": { "mime_prefix": "application/papermill.record/", "name": "fig_headvsU3_2" } }, "output_type": "display_data" }, { "data": { "image/svg+xml": [ "\n", "\n", "\n" ], "text/plain": [ "

" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# plot\n", "fig = plt.figure(figsize=diagrams.figdef1new)\n", "U = np.linspace(0, 1, 100)\n", "plt.plot(U, hmin(U), label='$h^-$')\n", "plt.plot(U, hplus(U), label='$h^+$')\n", "plt.xlabel('$U$ (m$^2$/d)')\n", "plt.ylabel('head (m) ')\n", "plt.grid()\n", "plt.legend()\n", "plt.tight_layout()\n", "\n", "glue(\"fig_headvsU3_2\", fig, display=False)" ] }, { "cell_type": "markdown", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "source": [ " ```{glue:figure} fig_headvsU3_2\n", "---\n", "name: fig_headvsU3.2\n", "---\n", "Head at $x=L_0$ vs. $U$.\n", "```" ] }, { "cell_type": "markdown", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "source": [ " As can be seen from the graph in {numref}`Figure %s `, the lines for $h^-$ and $h^+$ intersect somewhere between $U=0.6$ m$^2$/d and $U=0.8$ m$^2$/d. To determine the intersection point, a Python function is written to compute the difference between $h^-$ and $h^+$ for a given value of $U$. The value of $U$ for which this function is zero (so that $h^-=h^+$) is found with the `fsolve` method of the `scipy.optimize` package" ] }, { "cell_type": "code", "execution_count": 20, "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "U: 0.7139 m^2/d\n" ] } ], "source": [ "from scipy.optimize import fsolve\n", "\n", "def hdiff(U):\n", " return hmin(U) - hplus(U)\n", "\n", "U = fsolve(hdiff, 0.7)[0] # first value of array returned by fsolve\n", "print(f'U: {U:0.4f} m^2/d')" ] }, { "cell_type": "markdown", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "source": [ " The head can be computed with Eqs. ({eq}`eq_hneg_3.2`) and ({eq}`eq_hpos_3.2`) now that $U$ is known. The water table and streamlines in a vertical cross section are plotted in {numref}`Figure %s `. Note that the water table is continuous, but its slope (i.e., the head gradient) is not. This makes sense as the saturated thickness suddenly increases at the change in base while the discharge vector is continuous. This is only possible if at the same point the head gradient suddenly decreases. The elevations of the streamlines in {numref}`Figure %s ` jump at the change in the base. Both the jump in head gradient and the jumps in the streamlines are a consequence of the Dupuit–Forchheimer approximation, which means that streamlines can shift up or down without the loss of energy. In reality, the transition of both the head and the streamlines across a step in the base is smooth, as is demonstrated in {numref}`content:chap10:flow-over-step-base`." ] }, { "cell_type": "code", "execution_count": 23, "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "outputs": [], "source": [ "# solution\n", "x = np.hstack((np.linspace(0, L0 - 1e-6, 100), np.linspace(L0 + 1e-6, L, 100)))\n", "phi = np.empty_like(x)\n", "phi[x < L0] = -U * x[x < L0] + phil\n", "phi[x >= L0] = -U * (x[x >= L0] - L) + phir\n", "h = np.zeros_like(phi)\n", "h[x < L0] = np.sqrt(2 * phi[x < L0] / k) + z0\n", "h[x >= L0] = np.sqrt(2 * phi[x >= L0] / k) + z1\n", "#\n", "psi = np.zeros((2, len(x)))\n", "psi[1] = -U\n", "xg = np.zeros_like(psi)\n", "xg[:] = x\n", "zg = np.zeros_like(xg)\n", "zg[0, :100] = z0\n", "zg[0, 100:] = z1\n", "zg[1] = h" ] }, { "cell_type": "code", "execution_count": 25, "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [ "remove-output" ] }, "outputs": [ { "data": { "image/svg+xml": [ "\n", "\n", "\n" ], "text/plain": [ "

" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# basic streamline plot\n", "plt.subplot(111, aspect=25)\n", "plt.contour(xg, zg, psi, np.linspace(-U, 0, 4), colors='C1', linestyles='-')\n", "plt.plot(x, h, 'C0')\n", "plt.plot(x, zg[0], 'k');" ] }, { "cell_type": "code", "execution_count": 27, "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [ "remove-input", "remove-output" ] }, "outputs": [ { "data": { "application/papermill.record/image/svg+xml": "\n\n\n", "application/papermill.record/text/plain": "

" }, "metadata": { "scrapbook": { "mime_prefix": "application/papermill.record/", "name": "fig_stream3_2" } }, "output_type": "display_data" }, { "data": { "image/svg+xml": [ "\n", "\n", "\n" ], "text/plain": [ "

" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# plot\n", "fig = plt.figure(figsize=diagrams.figdef2)\n", "ax = plt.subplot(111)\n", "aspect = 20\n", "ax.set_aspect(aspect)\n", "plt.contour(xg, zg, psi, np.linspace(-U, 0, 4), colors='C1', linestyles='-')\n", "plt.plot(x, h, 'C0')\n", "plt.plot(x, zg[0], 'k')\n", "plt.xlabel('$x$ (m)')\n", "plt.ylabel(f'$z$ (m) - VE:{aspect}x')\n", "plt.tight_layout()\n", "\n", "glue(\"fig_stream3_2\", fig, display=False);" ] }, { "cell_type": "code", "execution_count": 29, "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [ "remove-input", "remove-output" ] }, "outputs": [ { "data": { "application/javascript": [ "'use strict';\n", "(function(root) {\n", " function now() {\n", " return new Date();\n", " }\n", "\n", " const force = true;\n", "\n", " if (typeof root._bokeh_onload_callbacks === \"undefined\" || force === true) {\n", " root._bokeh_onload_callbacks = [];\n", " root._bokeh_is_loading = undefined;\n", " }\n", "\n", "const JS_MIME_TYPE = 'application/javascript';\n", " const HTML_MIME_TYPE = 'text/html';\n", " const EXEC_MIME_TYPE = 'application/vnd.bokehjs_exec.v0+json';\n", " const CLASS_NAME = 'output_bokeh rendered_html';\n", "\n", " /**\n", " * Render data to the DOM node\n", " */\n", " function render(props, node) {\n", " const script = document.createElement(\"script\");\n", " node.appendChild(script);\n", " }\n", "\n", " /**\n", " * Handle when an output is cleared or removed\n", " */\n", " function handleClearOutput(event, handle) {\n", " function drop(id) {\n", " const view = Bokeh.index.get_by_id(id)\n", " if (view != null) {\n", " view.model.document.clear()\n", " Bokeh.index.delete(view)\n", " }\n", " }\n", "\n", " const cell = handle.cell;\n", "\n", " const id = cell.output_area._bokeh_element_id;\n", " const server_id = cell.output_area._bokeh_server_id;\n", "\n", " // Clean up Bokeh references\n", " if (id != null) {\n", " drop(id)\n", " }\n", "\n", " if (server_id !== undefined) {\n", " // Clean up Bokeh references\n", " const cmd_clean = \"from bokeh.io.state import curstate; 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\\n\"+\n", " \"

\"}};\n", "\n", " function display_loaded(error = null) {\n", " const el = document.getElementById(null);\n", " if (el != null) {\n", " const html = (() => {\n", " if (typeof root.Bokeh === \"undefined\") {\n", " if (error == null) {\n", " return \"BokehJS is loading ...\";\n", " } else {\n", " return \"BokehJS failed to load.\";\n", " }\n", " } else {\n", " const prefix = `BokehJS ${root.Bokeh.version}`;\n", " if (error == null) {\n", " return `${prefix} successfully loaded.`;\n", " } else {\n", " return `${prefix} encountered errors while loading and may not function as expected.`;\n", " }\n", " }\n", " })();\n", " el.innerHTML = html;\n", "\n", " if (error != null) {\n", " const wrapper = document.createElement(\"div\");\n", " wrapper.style.overflow = \"auto\";\n", " wrapper.style.height = \"5em\";\n", " wrapper.style.resize = \"vertical\";\n", " const content = document.createElement(\"div\");\n", " content.style.fontFamily = \"monospace\";\n", " content.style.whiteSpace = \"pre-wrap\";\n", " content.style.backgroundColor = \"rgb(255, 221, 221)\";\n", " content.textContent = error.stack ?? error.toString();\n", " wrapper.append(content);\n", " el.append(wrapper);\n", " }\n", " } else if (Date.now() < root._bokeh_timeout) {\n", " setTimeout(() => display_loaded(error), 100);\n", " }\n", " }\n", "\n", " function run_callbacks() {\n", " try {\n", " root._bokeh_onload_callbacks.forEach(function(callback) {\n", " if (callback != null)\n", " callback();\n", " });\n", " } finally {\n", " delete root._bokeh_onload_callbacks\n", " }\n", " console.debug(\"Bokeh: all callbacks have finished\");\n", " }\n", "\n", " function load_libs(css_urls, js_urls, callback) {\n", " if (css_urls == null) css_urls = [];\n", " if (js_urls == null) js_urls = [];\n", "\n", " root._bokeh_onload_callbacks.push(callback);\n", " if (root._bokeh_is_loading > 0) {\n", " console.debug(\"Bokeh: BokehJS is being loaded, scheduling callback at\", now());\n", " return null;\n", " }\n", " if (js_urls == null || js_urls.length === 0) {\n", " run_callbacks();\n", " return null;\n", " }\n", " console.debug(\"Bokeh: BokehJS not loaded, scheduling load and callback at\", now());\n", " root._bokeh_is_loading = css_urls.length + js_urls.length;\n", "\n", " function on_load() {\n", " root._bokeh_is_loading--;\n", " if (root._bokeh_is_loading === 0) {\n", " console.debug(\"Bokeh: all BokehJS libraries/stylesheets loaded\");\n", " run_callbacks()\n", " }\n", " }\n", "\n", " function on_error(url) {\n", " console.error(\"failed to load \" + url);\n", " }\n", "\n", " for (let i = 0; i < css_urls.length; i++) {\n", " const url = css_urls[i];\n", " const element = document.createElement(\"link\");\n", " element.onload = on_load;\n", " element.onerror = on_error.bind(null, url);\n", " element.rel = \"stylesheet\";\n", " element.type = \"text/css\";\n", " element.href = url;\n", " console.debug(\"Bokeh: injecting link tag for BokehJS stylesheet: \", url);\n", " document.body.appendChild(element);\n", " }\n", "\n", " for (let i = 0; i < js_urls.length; i++) {\n", " const url = js_urls[i];\n", " const element = document.createElement('script');\n", " element.onload = on_load;\n", " element.onerror = on_error.bind(null, url);\n", " element.async = false;\n", " element.src = url;\n", " console.debug(\"Bokeh: injecting script tag for BokehJS library: \", url);\n", " document.head.appendChild(element);\n", " }\n", " };\n", "\n", " function inject_raw_css(css) {\n", " const element = document.createElement(\"style\");\n", " element.appendChild(document.createTextNode(css));\n", " document.body.appendChild(element);\n", " }\n", "\n", " const js_urls = [\"https://cdn.bokeh.org/bokeh/release/bokeh-3.7.3.min.js\", \"https://cdn.bokeh.org/bokeh/release/bokeh-gl-3.7.3.min.js\", \"https://cdn.bokeh.org/bokeh/release/bokeh-widgets-3.7.3.min.js\", \"https://cdn.bokeh.org/bokeh/release/bokeh-tables-3.7.3.min.js\", \"https://cdn.bokeh.org/bokeh/release/bokeh-mathjax-3.7.3.min.js\"];\n", " const css_urls = [];\n", "\n", " const inline_js = [ function(Bokeh) {\n", " Bokeh.set_log_level(\"info\");\n", " },\n", "function(Bokeh) {\n", " }\n", " ];\n", "\n", " function run_inline_js() {\n", " if (root.Bokeh !== undefined || force === true) {\n", " try {\n", " for (let i = 0; i < inline_js.length; i++) {\n", " inline_js[i].call(root, root.Bokeh);\n", " }\n", "\n", " } catch (error) {throw error;\n", " }} else if (Date.now() < root._bokeh_timeout) {\n", " setTimeout(run_inline_js, 100);\n", " } else if (!root._bokeh_failed_load) {\n", " console.log(\"Bokeh: BokehJS failed to load within specified timeout.\");\n", " root._bokeh_failed_load = true;\n", " } else if (force !== true) {\n", " const cell = $(document.getElementById(null)).parents('.cell').data().cell;\n", " cell.output_area.append_execute_result(NB_LOAD_WARNING)\n", " }\n", " }\n", "\n", " if (root._bokeh_is_loading === 0) {\n", " console.debug(\"Bokeh: BokehJS loaded, going straight to plotting\");\n", " run_inline_js();\n", " } else {\n", " load_libs(css_urls, js_urls, function() {\n", " console.debug(\"Bokeh: BokehJS plotting callback run at\", now());\n", " run_inline_js();\n", " });\n", " }\n", "}(window));" ], "application/vnd.bokehjs_load.v0+json": "'use strict';\n(function(root) {\n function now() {\n return new Date();\n }\n\n const force = true;\n\n if (typeof root._bokeh_onload_callbacks === \"undefined\" || force === true) {\n root._bokeh_onload_callbacks = [];\n root._bokeh_is_loading = undefined;\n }\n\n\n if (typeof (root._bokeh_timeout) === \"undefined\" || force === true) {\n root._bokeh_timeout = Date.now() + 5000;\n root._bokeh_failed_load = false;\n }\n\n const NB_LOAD_WARNING = {'data': {'text/html':\n \"

\\n\"+\n \"

\\n\"+\n \"BokehJS does not appear to have successfully loaded. If loading BokehJS from CDN, this \\n\"+\n \"may be due to a slow or bad network connection. Possible fixes:\\n\"+\n \"

\\n\"+\n \"

re-rerun `output_notebook()` to attempt to load from CDN again, or
use INLINE resources instead, as so:

\\n\"+\n \"\\n\"+\n \"from bokeh.resources import INLINE\\n\"+\n \"output_notebook(resources=INLINE)\\n\"+\n \"\\n\"+\n \"

\"}};\n\n function display_loaded(error = null) {\n const el = document.getElementById(null);\n if (el != null) {\n const html = (() => {\n if (typeof root.Bokeh === \"undefined\") {\n if (error == null) {\n return \"BokehJS is loading ...\";\n } else {\n return \"BokehJS failed to load.\";\n }\n } else {\n const prefix = `BokehJS ${root.Bokeh.version}`;\n if (error == null) {\n return `${prefix} successfully loaded.`;\n } else {\n return `${prefix} encountered errors while loading and may not function as expected.`;\n }\n }\n })();\n el.innerHTML = html;\n\n if (error != null) {\n const wrapper = document.createElement(\"div\");\n wrapper.style.overflow = \"auto\";\n wrapper.style.height = \"5em\";\n wrapper.style.resize = \"vertical\";\n const content = document.createElement(\"div\");\n content.style.fontFamily = \"monospace\";\n content.style.whiteSpace = \"pre-wrap\";\n content.style.backgroundColor = \"rgb(255, 221, 221)\";\n content.textContent = error.stack ?? error.toString();\n wrapper.append(content);\n el.append(wrapper);\n }\n } else if (Date.now() < root._bokeh_timeout) {\n setTimeout(() => display_loaded(error), 100);\n }\n }\n\n function run_callbacks() {\n try {\n root._bokeh_onload_callbacks.forEach(function(callback) {\n if (callback != null)\n callback();\n });\n } finally {\n delete root._bokeh_onload_callbacks\n }\n console.debug(\"Bokeh: all callbacks have finished\");\n }\n\n function load_libs(css_urls, js_urls, callback) {\n if (css_urls == null) css_urls = [];\n if (js_urls == null) js_urls = [];\n\n root._bokeh_onload_callbacks.push(callback);\n if (root._bokeh_is_loading > 0) {\n console.debug(\"Bokeh: BokehJS is being loaded, scheduling callback at\", now());\n return null;\n }\n if (js_urls == null || js_urls.length === 0) {\n run_callbacks();\n return null;\n }\n console.debug(\"Bokeh: BokehJS not loaded, scheduling load and callback at\", now());\n root._bokeh_is_loading = css_urls.length + js_urls.length;\n\n function on_load() {\n root._bokeh_is_loading--;\n if (root._bokeh_is_loading === 0) {\n console.debug(\"Bokeh: all BokehJS libraries/stylesheets loaded\");\n run_callbacks()\n }\n }\n\n function on_error(url) {\n console.error(\"failed to load \" + url);\n }\n\n for (let i = 0; i < css_urls.length; i++) {\n const url = css_urls[i];\n const element = document.createElement(\"link\");\n element.onload = on_load;\n element.onerror = on_error.bind(null, url);\n element.rel = \"stylesheet\";\n element.type = \"text/css\";\n element.href = url;\n console.debug(\"Bokeh: injecting link tag for BokehJS stylesheet: \", url);\n document.body.appendChild(element);\n }\n\n for (let i = 0; i < js_urls.length; i++) {\n const url = js_urls[i];\n const element = document.createElement('script');\n element.onload = on_load;\n element.onerror = on_error.bind(null, url);\n element.async = false;\n element.src = url;\n console.debug(\"Bokeh: injecting script tag for BokehJS library: \", url);\n document.head.appendChild(element);\n }\n };\n\n function inject_raw_css(css) {\n const element = document.createElement(\"style\");\n element.appendChild(document.createTextNode(css));\n document.body.appendChild(element);\n }\n\n const js_urls = [\"https://cdn.bokeh.org/bokeh/release/bokeh-3.7.3.min.js\", \"https://cdn.bokeh.org/bokeh/release/bokeh-gl-3.7.3.min.js\", \"https://cdn.bokeh.org/bokeh/release/bokeh-widgets-3.7.3.min.js\", \"https://cdn.bokeh.org/bokeh/release/bokeh-tables-3.7.3.min.js\", \"https://cdn.bokeh.org/bokeh/release/bokeh-mathjax-3.7.3.min.js\"];\n const css_urls = [];\n\n const inline_js = [ function(Bokeh) {\n Bokeh.set_log_level(\"info\");\n },\nfunction(Bokeh) {\n }\n ];\n\n function run_inline_js() {\n if (root.Bokeh !== undefined || force === true) {\n try {\n for (let i = 0; i < inline_js.length; i++) {\n inline_js[i].call(root, root.Bokeh);\n }\n\n } catch (error) {throw error;\n }} else if (Date.now() < root._bokeh_timeout) {\n setTimeout(run_inline_js, 100);\n } else if (!root._bokeh_failed_load) {\n console.log(\"Bokeh: BokehJS failed to load within specified timeout.\");\n root._bokeh_failed_load = true;\n } else if (force !== true) {\n const cell = $(document.getElementById(null)).parents('.cell').data().cell;\n cell.output_area.append_execute_result(NB_LOAD_WARNING)\n }\n }\n\n if (root._bokeh_is_loading === 0) {\n console.debug(\"Bokeh: BokehJS loaded, going straight to plotting\");\n run_inline_js();\n } else {\n load_libs(css_urls, js_urls, function() {\n console.debug(\"Bokeh: BokehJS plotting callback run at\", now());\n run_inline_js();\n });\n }\n}(window));" }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/html": [ "\n", "

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const x = source.data['x'];\\n const h = source.data['h'];\\n const h0 = slider.value;\\n \\n const k = 10;\\n const z0 = 0;\\n const z1 = -4;\\n const L0 = 500;\\n const L1 = 500;\\n const L = L0 + L1;\\n const hL = 0;\\n \\n const phi0 = 0.5 * k * Math.pow(h0 - z0, 2);\\n const phiL = 0.5 * k * Math.pow(hL - z1, 2);\\n \\n function hmin(U) {\\n return Math.sqrt(2 * (-U * L0 + phi0) / k) + z0;\\n }\\n function hplus(U) {\\n return Math.sqrt(2 * (U * L1 + phiL) / k) + z1;\\n }\\n function hdiff(U) {\\n return hmin(U) - hplus(U);\\n }\\n \\n function findU() {\\n let U = 0.7;\\n for (let i = 0; i < 20; i++) {\\n let f = hdiff(U);\\n let df = (hdiff(U + 1e-6) - f) / 1e-6;\\n U = U - f / df;\\n }\\n return U;\\n }\\n \\n const U = findU();\\n \\n for (let i = 0; i < x.length; i++) {\\n let xi = x[i];\\n let phi;\\n if (xi < L0) {\\n phi = -U * xi + phi0;\\n h[i] = Math.sqrt(2 * phi / k) + z0;\\n } else {\\n phi = -U * (xi - L) + phiL;\\n h[i] = Math.sqrt(2 * phi / k) + z1;\\n }\\n }\\n \\n div_u.text = `U: ${U.toFixed(4)} m\\u00b2/d`;\\n \\n // Zoek juiste index van contour\\n let idx = slider_vals.findIndex(v => Math.abs(v - h0) < 1e-6);\\n if (idx === -1) return;\\n \\n // Update contourlijnen\\n const xs = all_xs[idx];\\n const ys = all_ys[idx];\\n \\n source2.data['x_cordinates'] = xs[0];\\n source2.data['y_cordinates'] = ys[0];\\n \\n source3.data['x_cordinates'] = xs[1];\\n source3.data['y_cordinates'] = ys[1];\\n \\n source4.data['x_cordinates'] = xs[2];\\n source4.data['y_cordinates'] = ys[2];\\n \\n source2.change.emit();\\n source3.change.emit();\\n source4.change.emit();\\n \"}}]]]},\"width\":570,\"margin\":10,\"align\":\"center\",\"title\":\"h0 (m)\",\"start\":5,\"end\":10,\"value\":10,\"step\":0.1}},{\"type\":\"object\",\"name\":\"Div\",\"id\":\"p1089\",\"attributes\":{\"align\":\"center\",\"text\":\"

5.00 m | 6.25 m | 7.50 m | 8.75 m | 10.00 m

\"}},{\"id\":\"p1090\"}]}}]}};\n", " const render_items = [{\"docid\":\"c4c4df9a-7fbd-4838-9d86-616182e529ac\",\"roots\":{\"p1093\":\"f049aa58-4198-4155-b3b7-72ac50dc7999\"},\"root_ids\":[\"p1093\"]}];\n", " void root.Bokeh.embed.embed_items_notebook(docs_json, render_items);\n", " }\n", " if (root.Bokeh !== undefined) {\n", " embed_document(root);\n", " } else {\n", " let attempts = 0;\n", " const timer = setInterval(function(root) {\n", " if (root.Bokeh !== undefined) {\n", " clearInterval(timer);\n", " embed_document(root);\n", " } else {\n", " attempts++;\n", " if (attempts > 100) {\n", " clearInterval(timer);\n", " console.log(\"Bokeh: ERROR: Unable to run BokehJS code because BokehJS library is missing\");\n", " }\n", " }\n", " }, 10, root)\n", " }\n", "})(window);" ], "application/vnd.bokehjs_exec.v0+json": "" }, "metadata": { "application/vnd.bokehjs_exec.v0+json": { "id": "p1093" } }, "output_type": "display_data" } ], "source": [ "inter_fig_chap3.fig_3_8()" ] }, { "cell_type": "markdown", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "source": [ "```{raw} html\n", "\n", "```\n", "\n", "```{figure} ../py_figures/1x1_pixel.png\n", "---\n", "name: fig_streamlines3.2\n", "---\n", "Streamlines for unconfined flow over a step in the base. Discharge between two streamlines is $\\Delta\\psi=U/3$.\n", "```" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "outputs": [], "source": [] } ], "metadata": { "celltoolbar": "Tags", "kernelspec": { "display_name": "Python 3 (ipykernel)", "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": "3.13.7" }, "latex_envs": { "LaTeX_envs_menu_present": true, "autoclose": false, "autocomplete": true, "bibliofile": "biblio.bib", "cite_by": "apalike", "current_citInitial": 1, "eqLabelWithNumbers": true, "eqNumInitial": 1, "hotkeys": { "equation": "Ctrl-E", "itemize": "Ctrl-I" }, "labels_anchors": false, "latex_user_defs": false, "report_style_numbering": false, "user_envs_cfg": false }, "varInspector": { "cols": { "lenName": 16, "lenType": 16, "lenVar": 40 }, "kernels_config": { "python": { "delete_cmd_postfix": "", "delete_cmd_prefix": "del ", "library": "var_list.py", "varRefreshCmd": "print(var_dic_list())" }, "r": { "delete_cmd_postfix": ") ", "delete_cmd_prefix": "rm(", "library": "var_list.r", "varRefreshCmd": "cat(var_dic_list()) " } }, "types_to_exclude": [ "module", "function", "builtin_function_or_method", "instance", "_Feature" ], "window_display": false } }, "nbformat": 4, "nbformat_minor": 4 }